Theorem gencbval 1837

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 groups:   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 610	. . . 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 1835	. . 3 |- (E.x(ch /\ -. ph) <-> E.y(th /\ -. ps))
7		exanali 1042	. . 3 |- (E.x(ch /\ -. ph) <-> -. A.x(ch -> ph))
8		exanali 1042	. . 3 |- (E.y(th /\ -. ps) <-> -. A.y(th -> ps))
9	6, 7, 8	3bitr3 181	. 2 |- (-. A.x(ch -> ph) <-> -. A.y(th -> ps))
10	9	con4bii 522	1 |- (A.x(ch -> ph) <-> A.y(th -> ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  E.wex 979  Vcvv 1808

This theorem is referenced by:  suppsr 5205  supsrlem6 5213  supre 5243

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809

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