Theorem gencbval 1373

Description: Change of bound variable using implicit substitution.

Hypotheses

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

Assertion

Ref	Expression
gencbval	|- (A.x(ch -> ph) <-> A.y(th -> ps))

Distinct variable group(s):   ps,x   ph,y   th,x   ch,y   y,A

Proof of Theorem gencbval

Step	Hyp	Ref	Expression
1		gencbval.1	. . . 4 |- A e. V
2		gencbval.2	. . . . 5 |- (A = y -> (ph <-> ps))
3	2	negbid 463	. . . 4 |- (A = y -> (-. ph <-> -. ps))
4		gencbval.3	. . . 4 |- (A = y -> (ch <-> th))
5		gencbval.4	. . . 4 |- (th <-> E.x(ch /\ A = y))
6	1, 3, 4, 5	gencbvex 1372	. . 3 |- (E.x(ch /\ -. ph) <-> E.y(th /\ -. ps))
7		exanali 725	. . 3 |- (E.x(ch /\ -. ph) <-> -. A.x(ch -> ph))
8		exanali 725	. . 3 |- (E.y(th /\ -. ps) <-> -. A.y(th -> ps))
9	6, 7, 8	3bitr3 156	. 2 |- (-. A.x(ch -> ph) <-> -. A.y(th -> ps))
10	9	bicon4i 401	1 |- (A.x(ch -> ph) <-> A.y(th -> ps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  suppsr 4016  supsrlem6 4024  supre 4054

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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