Theorem gencbval 3152

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 292	. . . 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 3150	. . 3 |- ( E. x ( ch /\ -. ph ) <-> E. y ( th /\ -. ps ) )
7		exanali 1675	. . 3 |- ( E. x ( ch /\ -. ph ) <-> -. A. x ( ch -> ph ) )
8		exanali 1675	. . 3 |- ( E. y ( th /\ -. ps ) <-> -. A. y ( th -> ps ) )
9	6, 7, 8	3bitr3i 275	. 2 |- ( -. A. x ( ch -> ph ) <-> -. A. y ( th -> ps ) )
10	9	con4bii 295	1 |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   A.wal 1396   = wceq 1398   E.wex 1617   e. wcel 1823   _Vcvv 3106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108

This theorem is referenced by: (None)