Theorem cbvsbcv 2984

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 1521	. 2 |- F/ y ph
2		nfv 1521	. 2 |- F/ x ps
3		cbvsbcv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvsbc 2983	1 |- ( [. A / x ]. ph <-> [. A / y ]. ps )

Syntax hints:   -> wi 4   <-> wb 104   [.wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-sbc 2956

This theorem is referenced by: (None)