Theorem cbvralv2 3203

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 2490	. 2 |- F/_yA
2		nfcv 2490	. 2 |- F/_xB
3		nfv 1619	. 2 |- F/yps
4		nfv 1619	. 2 |- F/xch
5		cbvralv2.2	. 2 |- (x = y -> A = B)
6		cbvralv2.1	. 2 |- (x = y -> (ps <-> ch))
7	1, 2, 3, 4, 5, 6	cbvralcsf 3199	1 |- (A.x e. A ps <-> A.y e. B ch)

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)