Theorem gencbval 2667

Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)

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		alcom 1412	. 2 |- ( A. x A. y ( y = A -> ( th -> ps ) ) <-> A. y A. x ( y = A -> ( th -> ps ) ) )
2		gencbval.1	. . . 4 |- A e. _V
3		gencbval.3	. . . . . . 7 |- ( A = y -> ( ch <-> th ) )
4		gencbval.2	. . . . . . 7 |- ( A = y -> ( ph <-> ps ) )
5	3, 4	imbi12d 232	. . . . . 6 |- ( A = y -> ( ( ch -> ph ) <-> ( th -> ps ) ) )
6	5	bicomd 139	. . . . 5 |- ( A = y -> ( ( th -> ps ) <-> ( ch -> ph ) ) )
7	6	eqcoms 2091	. . . 4 |- ( y = A -> ( ( th -> ps ) <-> ( ch -> ph ) ) )
8	2, 7	ceqsalv 2649	. . 3 |- ( A. y ( y = A -> ( th -> ps ) ) <-> ( ch -> ph ) )
9	8	albii 1404	. 2 |- ( A. x A. y ( y = A -> ( th -> ps ) ) <-> A. x ( ch -> ph ) )
10		19.23v 1811	. . . 4 |- ( A. x ( y = A -> ( th -> ps ) ) <-> ( E. x y = A -> ( th -> ps ) ) )
11		gencbval.4	. . . . . . 7 |- ( th <-> E. x ( ch /\ A = y ) )
12		eqcom 2090	. . . . . . . . . 10 |- ( A = y <-> y = A )
13	12	biimpi 118	. . . . . . . . 9 |- ( A = y -> y = A )
14	13	adantl 271	. . . . . . . 8 |- ( ( ch /\ A = y ) -> y = A )
15	14	eximi 1536	. . . . . . 7 |- ( E. x ( ch /\ A = y ) -> E. x y = A )
16	11, 15	sylbi 119	. . . . . 6 |- ( th -> E. x y = A )
17		pm2.04 81	. . . . . 6 |- ( ( E. x y = A -> ( th -> ps ) ) -> ( th -> ( E. x y = A -> ps ) ) )
18	16, 17	mpdi 42	. . . . 5 |- ( ( E. x y = A -> ( th -> ps ) ) -> ( th -> ps ) )
19		ax-1 5	. . . . 5 |- ( ( th -> ps ) -> ( E. x y = A -> ( th -> ps ) ) )
20	18, 19	impbii 124	. . . 4 |- ( ( E. x y = A -> ( th -> ps ) ) <-> ( th -> ps ) )
21	10, 20	bitri 182	. . 3 |- ( A. x ( y = A -> ( th -> ps ) ) <-> ( th -> ps ) )
22	21	albii 1404	. 2 |- ( A. y A. x ( y = A -> ( th -> ps ) ) <-> A. y ( th -> ps ) )
23	1, 9, 22	3bitr3i 208	1 |- ( A. x ( ch -> ph ) <-> A. y ( th -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1287   = wceq 1289   E.wex 1426   e. wcel 1438   _Vcvv 2619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621

This theorem is referenced by: (None)