Theorem gencbval 2904

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)

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

Allowed substitution hints:   ph(x)   ps(y)   ch(x)   th(y)   A(x)

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	notbid 285	. . . 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 2902	. . 3 |- (E.x(ch /\ -. ph) <-> E.y(th /\ -. ps))
7		exanali 1585	. . 3 |- (E.x(ch /\ -. ph) <-> -. A.x(ch -> ph))
8		exanali 1585	. . 3 |- (E.y(th /\ -. ps) <-> -. A.y(th -> ps))
9	6, 7, 8	3bitr3i 266	. 2 |- (-. A.x(ch -> ph) <-> -. A.y(th -> ps))
10	9	con4bii 288	1 |- (A.x(ch -> ph) <-> A.y(th -> ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  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: (None)