Theorem cbvex2 2005

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
cbval2.1	|- F/zph
cbval2.2	|- F/wph
cbval2.3	|- F/xps
cbval2.4	|- F/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

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

Proof of Theorem cbvex2

Step	Hyp	Ref	Expression
1		cbval2.1	. . 3 |- F/zph
2	1	nfex 1843	. 2 |- F/zE.yph
3		cbval2.3	. . 3 |- F/xps
4	3	nfex 1843	. 2 |- F/xE.wps
5		nfv 1619	. . . . . 6 |- F/w x = z
6		cbval2.2	. . . . . 6 |- F/wph
7	5, 6	nfan 1824	. . . . 5 |- F/w(x = z /\ ph)
8		nfv 1619	. . . . . 6 |- F/y x = z
9		cbval2.4	. . . . . 6 |- F/yps
10	8, 9	nfan 1824	. . . . 5 |- F/y(x = z /\ ps)
11		cbval2.5	. . . . . . 7 |- ((x = z /\ y = w) -> (ph <-> ps))
12	11	expcom 424	. . . . . 6 |- (y = w -> (x = z -> (ph <-> ps)))
13	12	pm5.32d 620	. . . . 5 |- (y = w -> ((x = z /\ ph) <-> (x = z /\ ps)))
14	7, 10, 13	cbvex 1985	. . . 4 |- (E.y(x = z /\ ph) <-> E.w(x = z /\ ps))
15		19.42v 1905	. . . 4 |- (E.y(x = z /\ ph) <-> (x = z /\ E.yph))
16		19.42v 1905	. . . 4 |- (E.w(x = z /\ ps) <-> (x = z /\ E.wps))
17	14, 15, 16	3bitr3i 266	. . 3 |- ((x = z /\ E.yph) <-> (x = z /\ E.wps))
18		pm5.32 617	. . 3 |- ((x = z -> (E.yph <-> E.wps)) <-> ((x = z /\ E.yph) <-> (x = z /\ E.wps)))
19	17, 18	mpbir 200	. 2 |- (x = z -> (E.yph <-> E.wps))
20	2, 4, 19	cbvex 1985	1 |- (E.xE.yph <-> E.zE.wps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  cbvex2v  2007  2eu6  2289  cbvopab  4631  cbvoprab12  5570