Theorem cbvralv2 3061

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
cbvralv2.1	|- ( x = y -> ( ps <-> ch ) )
cbvralv2.2	|- ( x = y -> A = B )

Assertion

Ref	Expression
cbvralv2	|- ( A. x e. A ps <-> A. y e. B ch )

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

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

Proof of Theorem cbvralv2

Step	Hyp	Ref	Expression
1		nfcv 2279	. 2 |- F/_ y A
2		nfcv 2279	. 2 |- F/_ x B
3		nfv 1508	. 2 |- F/ y ps
4		nfv 1508	. 2 |- F/ x ch
5		cbvralv2.2	. 2 |- ( x = y -> A = B )
6		cbvralv2.1	. 2 |- ( x = y -> ( ps <-> ch ) )
7	1, 2, 3, 4, 5, 6	cbvralcsf 3057	1 |- ( A. x e. A ps <-> A. y e. B ch )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   A.wral 2414

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-sbc 2905  df-csb 2999

This theorem is referenced by: (None)