Theorem gencbvex 2902

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 1741	. 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 691	. . . . . 6 |- (A = y -> ((ch /\ ph) <-> (th /\ ps)))
6	5	bicomd 192	. . . . 5 |- (A = y -> ((th /\ ps) <-> (ch /\ ph)))
7	6	eqcoms 2356	. . . 4 |- (y = A -> ((th /\ ps) <-> (ch /\ ph)))
8	2, 7	ceqsexv 2895	. . 3 |- (E.y(y = A /\ (th /\ ps)) <-> (ch /\ ph))
9	8	exbii 1582	. 2 |- (E.xE.y(y = A /\ (th /\ ps)) <-> E.x(ch /\ ph))
10		19.41v 1901	. . . 4 |- (E.x(y = A /\ (th /\ ps)) <-> (E.x y = A /\ (th /\ ps)))
11		simpr 447	. . . . 5 |- ((E.x y = A /\ (th /\ ps)) -> (th /\ ps))
12		gencbvex.4	. . . . . . . 8 |- (th <-> E.x(ch /\ A = y))
13		eqcom 2355	. . . . . . . . . . 11 |- (A = y <-> y = A)
14	13	biimpi 186	. . . . . . . . . 10 |- (A = y -> y = A)
15	14	adantl 452	. . . . . . . . 9 |- ((ch /\ A = y) -> y = A)
16	15	eximi 1576	. . . . . . . 8 |- (E.x(ch /\ A = y) -> E.x y = A)
17	12, 16	sylbi 187	. . . . . . 7 |- (th -> E.x y = A)
18	17	adantr 451	. . . . . 6 |- ((th /\ ps) -> E.x y = A)
19	18	ancri 535	. . . . 5 |- ((th /\ ps) -> (E.x y = A /\ (th /\ ps)))
20	11, 19	impbii 180	. . . 4 |- ((E.x y = A /\ (th /\ ps)) <-> (th /\ ps))
21	10, 20	bitri 240	. . 3 |- (E.x(y = A /\ (th /\ ps)) <-> (th /\ ps))
22	21	exbii 1582	. 2 |- (E.yE.x(y = A /\ (th /\ ps)) <-> E.y(th /\ ps))
23	1, 9, 22	3bitr3i 266	1 |- (E.x(ch /\ ph) <-> E.y(th /\ ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860

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-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  gencbvex2  2903  gencbval  2904