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Step * of Lemma uorder_functionality_wrt_iff

∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ].
  ((∀[x,y:T].  (R[x;y] ⇐⇒ R'[x;y])) ⇒ (UniformOrder(T;x,y.R[x;y]) ⇐⇒ UniformOrder(T;x,y.R'[x;y])))
BY
{ (Auto THEN ParallelLast) }

1
1. [T] : Type
2. [R] : T ⟶ T ⟶ ℙ
3. [R'] : T ⟶ T ⟶ ℙ
4. ∀[x,y:T].  (R[x;y] ⇐⇒ R'[x;y])
5. UniformlyRefl(T;x,y.R[x;y]) ∧ UniformlyTrans(T;x,y.R[x;y]) ∧ UniformlyAntiSym(T;x,y.R[x;y])
⊢ UniformlyRefl(T;x,y.R'[x;y]) ∧ UniformlyTrans(T;x,y.R'[x;y]) ∧ UniformlyAntiSym(T;x,y.R'[x;y])

2
1. [T] : Type
2. [R] : T ⟶ T ⟶ ℙ
3. [R'] : T ⟶ T ⟶ ℙ
4. ∀[x,y:T].  (R[x;y] ⇐⇒ R'[x;y])
5. UniformlyRefl(T;x,y.R'[x;y]) ∧ UniformlyTrans(T;x,y.R'[x;y]) ∧ UniformlyAntiSym(T;x,y.R'[x;y])
⊢ UniformlyRefl(T;x,y.R[x;y]) ∧ UniformlyTrans(T;x,y.R[x;y]) ∧ UniformlyAntiSym(T;x,y.R[x;y])

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Latex:

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Latex:
\mforall{}[T:Type].  \mforall{}[R,R':T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
    ((\mforall{}[x,y:T].    (R[x;y]  \mLeftarrow{}{}\mRightarrow{}  R'[x;y]))
    {}\mRightarrow{}  (UniformOrder(T;x,y.R[x;y])  \mLeftarrow{}{}\mRightarrow{}  UniformOrder(T;x,y.R'[x;y])))


By

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Latex:
(Auto  THEN  ParallelLast)

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