Theorem gencbval 3252

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 308	. . . 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 3250	. . 3 |- ( E. x ( ch /\ -. ph ) <-> E. y ( th /\ -. ps ) )
7		exanali 1786	. . 3 |- ( E. x ( ch /\ -. ph ) <-> -. A. x ( ch -> ph ) )
8		exanali 1786	. . 3 |- ( E. y ( th /\ -. ps ) <-> -. A. y ( th -> ps ) )
9	6, 7, 8	3bitr3i 290	. 2 |- ( -. A. x ( ch -> ph ) <-> -. A. y ( th -> ps ) )
10	9	con4bii 311	1 |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-11 2034  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by: (None)