Theorem cbvex2 975

Description: Rule used to change bound variables with implicit substitution.

Hypotheses

Ref	Expression
cbval2.1	|- (ph -> A.zph)
cbval2.2	|- (ph -> A.wph)
cbval2.3	|- (ps -> A.xps)
cbval2.4	|- (ps -> A.yps)
cbval2.5	|- ((x = z /\ y = w) -> (ph <-> ps))

Assertion

Ref	Expression
cbvex2	|- (E.xE.yph <-> E.zE.wps)

Distinct variable group(s):   x,y   y,z   x,w   z,w

Proof of Theorem cbvex2

Step	Hyp	Ref	Expression
1		cbval2.1	. . 3 |- (ph -> A.zph)
2	1	hbex 701	. 2 |- (E.yph -> A.zE.yph)
3		cbval2.3	. . 3 |- (ps -> A.xps)
4	3	hbex 701	. 2 |- (E.wps -> A.xE.wps)
5		ax-17 925	. . . . . 6 |- (x = z -> A.w x = z)
6		cbval2.2	. . . . . 6 |- (ph -> A.wph)
7	5, 6	hban 704	. . . . 5 |- ((x = z /\ ph) -> A.w(x = z /\ ph))
8		ax-17 925	. . . . . 6 |- (x = z -> A.y x = z)
9		cbval2.4	. . . . . 6 |- (ps -> A.yps)
10	8, 9	hban 704	. . . . 5 |- ((x = z /\ ps) -> A.y(x = z /\ ps))
11		cbval2.5	. . . . . . . 8 |- ((x = z /\ y = w) -> (ph <-> ps))
12	11	exp 291	. . . . . . 7 |- (x = z -> (y = w -> (ph <-> ps)))
13	12	com12 13	. . . . . 6 |- (y = w -> (x = z -> (ph <-> ps)))
14	13	pm5.32d 491	. . . . 5 |- (y = w -> ((x = z /\ ph) <-> (x = z /\ ps)))
15	7, 10, 14	cbvex 849	. . . 4 |- (E.y(x = z /\ ph) <-> E.w(x = z /\ ps))
16	8	19.42 775	. . . 4 |- (E.y(x = z /\ ph) <-> (x = z /\ E.yph))
17	5	19.42 775	. . . 4 |- (E.w(x = z /\ ps) <-> (x = z /\ E.wps))
18	15, 16, 17	3bitr3 156	. . 3 |- ((x = z /\ E.yph) <-> (x = z /\ E.wps))
19		pm5.32 488	. . 3 |- ((x = z -> (E.yph <-> E.wps)) <-> ((x = z /\ E.yph) <-> (x = z /\ E.wps)))
20	18, 19	mpbir 165	. 2 |- (x = z -> (E.yph <-> E.wps))
21	2, 4, 20	cbvex 849	1 |- (E.xE.yph <-> E.zE.wps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797

This theorem is referenced by:  cbvex2v 976  cbvopab 2104  cbvoprab12 3028

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

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