Theorem gencbvex 2334

Description: Change of bound variable using implicit substitution. (The proof was 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

Proof of Theorem gencbvex

Step	Hyp	Ref	Expression
1		excom 1393	. 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 690	. . . . . 6 |- (A = y -> ((ch /\ ph) <-> (th /\ ps)))
6	5	bicomd 580	. . . . 5 |- (A = y -> ((th /\ ps) <-> (ch /\ ph)))
7	6	eqcoms 1887	. . . 4 |- (y = A -> ((th /\ ps) <-> (ch /\ ph)))
8	2, 7	ceqsexv 2325	. . 3 |- (E.y(y = A /\ (th /\ ps)) <-> (ch /\ ph))
9	8	exbii 1398	. 2 |- (E.xE.y(y = A /\ (th /\ ps)) <-> E.x(ch /\ ph))
10		19.41v 1685	. . . 4 |- (E.x(y = A /\ (th /\ ps)) <-> (E.x y = A /\ (th /\ ps)))
11		simpr 350	. . . . 5 |- ((E.x y = A /\ (th /\ ps)) -> (th /\ ps))
12		gencbvex.4	. . . . . . . 8 |- (th <-> E.x(ch /\ A = y))
13		eqcom 1886	. . . . . . . . . . 11 |- (A = y <-> y = A)
14	13	biimpi 168	. . . . . . . . . 10 |- (A = y -> y = A)
15	14	adantl 424	. . . . . . . . 9 |- ((ch /\ A = y) -> y = A)
16	15	eximi 1387	. . . . . . . 8 |- (E.x(ch /\ A = y) -> E.x y = A)
17	12, 16	sylbi 216	. . . . . . 7 |- (th -> E.x y = A)
18	17	adantr 425	. . . . . 6 |- ((th /\ ps) -> E.x y = A)
19	18	ancri 321	. . . . 5 |- ((th /\ ps) -> (E.x y = A /\ (th /\ ps)))
20	11, 19	impbii 174	. . . 4 |- ((E.x y = A /\ (th /\ ps)) <-> (th /\ ps))
21	10, 20	bitri 190	. . 3 |- (E.x(y = A /\ (th /\ ps)) <-> (th /\ ps))
22	21	exbii 1398	. 2 |- (E.yE.x(y = A /\ (th /\ ps)) <-> E.y(th /\ ps))
23	1, 9, 22	3bitr3i 198	1 |- (E.x(ch /\ ph) <-> E.y(th /\ ps))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292

This theorem is referenced by:  gencbvex2 2336  gencbval 2337  suppsr 6374  supsrlem6 6382  supre 6412

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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