Theorem cbvral2v 1801

Description: Change bound variables of double restricted universal quantification, using implicit substitution.

Hypotheses

Ref	Expression
cbvral2v.1	|- (x = z -> (ph <-> ch))
cbvral2v.2	|- (y = w -> (ch <-> ps))

Assertion

Ref	Expression
cbvral2v	|- (A.x e. A A.y e. B ph <-> A.z e. A A.w e. B ps)

Distinct variable groups:   x,z,A   x,y,B,z   y,w,B   ph,z   ps,y   ch,x   ch,w

Proof of Theorem cbvral2v

Step	Hyp	Ref	Expression
1		cbvral2v.1	. . . 4 |- (x = z -> (ph <-> ch))
2	1	ralbidv 1662	. . 3 |- (x = z -> (A.y e. B ph <-> A.y e. B ch))
3	2	cbvralv 1798	. 2 |- (A.x e. A A.y e. B ph <-> A.z e. A A.y e. B ch)
4		cbvral2v.2	. . . 4 |- (y = w -> (ch <-> ps))
5	4	cbvralv 1798	. . 3 |- (A.y e. B ch <-> A.w e. B ps)
6	5	ralbii 1666	. 2 |- (A.z e. A A.y e. B ch <-> A.z e. A A.w e. B ps)
7	3, 6	bitr 173	1 |- (A.x e. A A.y e. B ph <-> A.z e. A A.w e. B ps)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 955  A.wral 1644

This theorem is referenced by:  cbvral3v 1802  fununi 3560  fiint 4547  ghgrpilem1 8118  cdj3lem3b 10358  cdj3 10359  cayleylem2 10401

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-cleq 1469  df-clel 1472  df-ral 1648

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