Theorem gencbvex 2594

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
gencbvex.1	|- A e. _V
gencbvex.2	|- (A = y -> (ph <-> ps))
gencbvex.3	|- (A = y -> (ch <-> th))
gencbvex.4	|- (th <-> E.x(ch /\ A = y))

Assertion

Ref	Expression
gencbvex	|- (E.x(ch /\ ph) <-> E.y(th /\ ps))

Distinct variable groups:   ps,x   ph,y   th,x   ch,y   y,A

Allowed substitution hints:   ph(x)   ps(y)   ch(x)   th(y)   A(x)

Proof of Theorem gencbvex

Step	Hyp	Ref	Expression
1		excom 1551	. 2 |- (E.xE.y(y = A /\ (th /\ ps)) <-> E.yE.x(y = A /\ (th /\ ps)))
2		gencbvex.1	. . . 4 |- A e. _V
3		gencbvex.3	. . . . . . 7 |- (A = y -> (ch <-> th))
4		gencbvex.2	. . . . . . 7 |- (A = y -> (ph <-> ps))
5	3, 4	anbi12d 442	. . . . . 6 |- (A = y -> ((ch /\ ph) <-> (th /\ ps)))
6	5	bicomd 129	. . . . 5 |- (A = y -> ((th /\ ps) <-> (ch /\ ph)))
7	6	eqcoms 2040	. . . 4 |- (y = A -> ((th /\ ps) <-> (ch /\ ph)))
8	2, 7	ceqsexv 2587	. . 3 |- (E.y(y = A /\ (th /\ ps)) <-> (ch /\ ph))
9	8	exbii 1493	. 2 |- (E.xE.y(y = A /\ (th /\ ps)) <-> E.x(ch /\ ph))
10		19.41v 1779	. . . 4 |- (E.x(y = A /\ (th /\ ps)) <-> (E.x y = A /\ (th /\ ps)))
11		simpr 103	. . . . 5 |- ((E.x y = A /\ (th /\ ps)) -> (th /\ ps))
12		gencbvex.4	. . . . . . . 8 |- (th <-> E.x(ch /\ A = y))
13		eqcom 2039	. . . . . . . . . . 11 |- (A = y <-> y = A)
14	13	biimpi 113	. . . . . . . . . 10 |- (A = y -> y = A)
15	14	adantl 262	. . . . . . . . 9 |- ((ch /\ A = y) -> y = A)
16	15	eximi 1488	. . . . . . . 8 |- (E.x(ch /\ A = y) -> E.x y = A)
17	12, 16	sylbi 114	. . . . . . 7 |- (th -> E.x y = A)
18	17	adantr 261	. . . . . 6 |- ((th /\ ps) -> E.x y = A)
19	18	ancri 307	. . . . 5 |- ((th /\ ps) -> (E.x y = A /\ (th /\ ps)))
20	11, 19	impbii 117	. . . 4 |- ((E.x y = A /\ (th /\ ps)) <-> (th /\ ps))
21	10, 20	bitri 173	. . 3 |- (E.x(y = A /\ (th /\ ps)) <-> (th /\ ps))
22	21	exbii 1493	. 2 |- (E.yE.x(y = A /\ (th /\ ps)) <-> E.y(th /\ ps))
23	1, 9, 22	3bitr3i 199	1 |- (E.x(ch /\ ph) <-> E.y(th /\ ps))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  gencbvex2  2595