Theorem cbvral2v 2844

Description: Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)

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,A   z,A   x,y,B   y,z,B   w,B   ph,z   ps,y   ch,x   ch,w

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

Proof of Theorem cbvral2v

Step	Hyp	Ref	Expression
1		cbvral2v.1	. . . 4 |- (x = z -> (ph <-> ch))
2	1	ralbidv 2635	. . 3 |- (x = z -> (A.y e. B ph <-> A.y e. B ch))
3	2	cbvralv 2836	. 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 2836	. . 3 |- (A.y e. B ch <-> A.w e. B ps)
6	5	ralbii 2639	. 2 |- (A.z e. A A.y e. B ch <-> A.z e. A A.w e. B ps)
7	3, 6	bitri 240	1 |- (A.x e. A A.y e. B ph <-> A.z e. A A.w e. B ps)

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-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  cbvral3v  2846  nnpweq  4524  fununi  5161