Theorem cbvabv 1424

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

Hypothesis

Ref	Expression
cbvabv.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvabv	⊢ {x∣φ} = {y∣ψ}

Distinct variable group(s):   x,y   φ,y   ψ,x

Proof of Theorem cbvabv

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (φ → ∀yφ)
2		ax-17 925	. 2 ⊢ (ψ → ∀xψ)
3		cbvabv.1	. 2 ⊢ (x = y → (φ ↔ ψ))
4	1, 2, 3	cbvab 1423	1 ⊢ {x∣φ} = {y∣ψ}

Syntax hints:   → wi 2   ↔ wb 127   = weq 797  {cab 1090   = wceq 1091

This theorem is referenced by:  sbth 3359  aceq3lem 3555  zorn 3611  genpv 3896  ltexpri 3943  recexpr 3954  suppsr2 4017  supsrlem4 4022  supsrlem6 4024  supsr 4025  axsup 4088  infmap2lem1 4951  ch2 5149

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-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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