Theorem gencbvex 1838

Description: Change of bound variable using implicit substitution.

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 1046	. 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 628	. . . . . 6 |- (A = y -> ((ch /\ ph) <-> (th /\ ps)))
6	5	bicomd 521	. . . . 5 |- (A = y -> ((th /\ ps) <-> (ch /\ ph)))
7	6	eqcoms 1478	. . . 4 |- (y = A -> ((th /\ ps) <-> (ch /\ ph)))
8	2, 7	ceqsexv 1835	. . 3 |- (E.y(y = A /\ (th /\ ps)) <-> (ch /\ ph))
9	8	exbii 1051	. 2 |- (E.xE.y(y = A /\ (th /\ ps)) <-> E.x(ch /\ ph))
10		anass 439	. . . 4 |- (((E.x y = A /\ th) /\ ps) <-> (E.x y = A /\ (th /\ ps)))
11		gencbvex.4	. . . . . 6 |- (th <-> E.x(ch /\ A = y))
12	3	pm5.32i 645	. . . . . . . 8 |- ((A = y /\ ch) <-> (A = y /\ th))
13		ancom 435	. . . . . . . 8 |- ((A = y /\ ch) <-> (ch /\ A = y))
14		eqcom 1477	. . . . . . . . 9 |- (A = y <-> y = A)
15	14	anbi1i 481	. . . . . . . 8 |- ((A = y /\ th) <-> (y = A /\ th))
16	12, 13, 15	3bitr3 181	. . . . . . 7 |- ((ch /\ A = y) <-> (y = A /\ th))
17	16	exbii 1051	. . . . . 6 |- (E.x(ch /\ A = y) <-> E.x(y = A /\ th))
18		19.41v 1305	. . . . . 6 |- (E.x(y = A /\ th) <-> (E.x y = A /\ th))
19	11, 17, 18	3bitr 177	. . . . 5 |- (th <-> (E.x y = A /\ th))
20	19	anbi1i 481	. . . 4 |- ((th /\ ps) <-> ((E.x y = A /\ th) /\ ps))
21		19.41v 1305	. . . 4 |- (E.x(y = A /\ (th /\ ps)) <-> (E.x y = A /\ (th /\ ps)))
22	10, 20, 21	3bitr4r 184	. . 3 |- (E.x(y = A /\ (th /\ ps)) <-> (th /\ ps))
23	22	exbii 1051	. 2 |- (E.yE.x(y = A /\ (th /\ ps)) <-> E.y(th /\ ps))
24	1, 9, 23	3bitr3 181	1 |- (E.x(ch /\ ph) <-> E.y(th /\ ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811

This theorem is referenced by:  gencbvex2 1839  gencbval 1840  suppsr 5222  supsrlem6 5230  supre 5260

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

Copyright terms: Public domain