Theorem cbvrabv2 39311

Description: A more general version of cbvrabv 3199. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
cbvrabv2.1	|- ( x = y -> A = B )
cbvrabv2.2	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvrabv2	|- { x e. A | ph } = { y e. B | ps }

Distinct variable groups:   y, A   x, B   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)   A( x)   B( y)

Proof of Theorem cbvrabv2

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ y A
2		nfcv 2764	. 2 |- F/_ x B
3		nfv 1843	. 2 |- F/ y ph
4		nfv 1843	. 2 |- F/ x ps
5		cbvrabv2.1	. 2 |- ( x = y -> A = B )
6		cbvrabv2.2	. 2 |- ( x = y -> ( ph <-> ps ) )
7	1, 2, 3, 4, 5, 6	cbvrabcsf 3568	1 |- { x e. A | ph } = { y e. B | ps }

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-sbc 3436  df-csb 3534

This theorem is referenced by:  smfsuplem2  41018