Theorem cbvex2 1319

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 groups:   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 1008	. 2 |- (E.yph -> A.zE.yph)
3		cbval2.3	. . 3 |- (ps -> A.xps)
4	3	hbex 1008	. 2 |- (E.wps -> A.xE.wps)
5		ax-17 973	. . . . . 6 |- (x = z -> A.w x = z)
6		cbval2.2	. . . . . 6 |- (ph -> A.wph)
7	5, 6	hban 1011	. . . . 5 |- ((x = z /\ ph) -> A.w(x = z /\ ph))
8		ax-17 973	. . . . . 6 |- (x = z -> A.y x = z)
9		cbval2.4	. . . . . 6 |- (ps -> A.yps)
10	8, 9	hban 1011	. . . . 5 |- ((x = z /\ ps) -> A.y(x = z /\ ps))
11		cbval2.5	. . . . . . 7 |- ((x = z /\ y = w) -> (ph <-> ps))
12	11	expcom 374	. . . . . 6 |- (y = w -> (x = z -> (ph <-> ps)))
13	12	pm5.32d 649	. . . . 5 |- (y = w -> ((x = z /\ ph) <-> (x = z /\ ps)))
14	7, 10, 13	cbvex 1168	. . . 4 |- (E.y(x = z /\ ph) <-> E.w(x = z /\ ps))
15	8	19.42 1098	. . . 4 |- (E.y(x = z /\ ph) <-> (x = z /\ E.yph))
16	5	19.42 1098	. . . 4 |- (E.w(x = z /\ ps) <-> (x = z /\ E.wps))
17	14, 15, 16	3bitr3 181	. . 3 |- ((x = z /\ E.yph) <-> (x = z /\ E.wps))
18		pm5.32 646	. . 3 |- ((x = z -> (E.yph <-> E.wps)) <-> ((x = z /\ E.yph) <-> (x = z /\ E.wps)))
19	17, 18	mpbir 190	. 2 |- (x = z -> (E.yph <-> E.wps))
20	2, 4, 19	cbvex 1168	1 |- (E.xE.yph <-> E.zE.wps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958  E.wex 982

This theorem is referenced by:  cbvex2v 1321  2eu6 1457  cbvopab 2677  cbvoprab12 4004

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983

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