Theorem cbvsbcv 3004

Description: Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
cbvsbcv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvsbcv	|- ( [. A / x ]. ph <-> [. A / y ]. ps )

Distinct variable groups:   ph, y   ps, x

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

Proof of Theorem cbvsbcv

Step	Hyp	Ref	Expression
1		nfv 1538	. 2 |- F/ y ph
2		nfv 1538	. 2 |- F/ x ps
3		cbvsbcv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvsbc 3003	1 |- ( [. A / x ]. ph <-> [. A / y ]. ps )

Syntax hints:   -> wi 4   <-> wb 105   [.wsbc 2974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-sbc 2975

This theorem is referenced by: (None)