Theorem cbvral 1331

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbvral.1	⊢ (φ → ∀yφ)
cbvral.2	⊢ (ψ → ∀xψ)
cbvral.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvral	⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)

Distinct variable group(s):   x,y,A

Proof of Theorem cbvral

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (z ∈ A → ∀x z ∈ A)
2		ax-17 925	. 2 ⊢ (z ∈ A → ∀y z ∈ A)
3		cbvral.1	. 2 ⊢ (φ → ∀yφ)
4		cbvral.2	. 2 ⊢ (ψ → ∀xψ)
5		cbvral.3	. 2 ⊢ (x = y → (φ ↔ ψ))
6	1, 2, 3, 4, 5	cbvralf 1330	1 ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = weq 797   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  cbvralv 1333  sbralie 1439  tfinds 2401   2404  cleqfvf 2881  f1fvf 2917  >A HREF="tfrlem1.html">tfrlem1 2949

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ral 1205

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