Theorem ceqex 2671

Description: Equality implies equivalence with substitution. (Contributed by NM, 2-Mar-1995.)

Assertion

Ref	Expression
ceqex	|- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem ceqex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.8a 1482	. . 3 |- ( x = A -> E. x x = A )
2		isset 2561	. . 3 |- ( A e. _V <-> E. x x = A )
3	1, 2	sylibr 137	. 2 |- ( x = A -> A e. _V )
4		eqeq2 2049	. . . 4 |- ( y = A -> ( x = y <-> x = A ) )
5	4	anbi1d 438	. . . . . 6 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
6	5	exbidv 1706	. . . . 5 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
7	6	bibi2d 221	. . . 4 |- ( y = A -> ( ( ph <-> E. x ( x = y /\ ph ) ) <-> ( ph <-> E. x ( x = A /\ ph ) ) ) )
8	4, 7	imbi12d 223	. . 3 |- ( y = A -> ( ( x = y -> ( ph <-> E. x ( x = y /\ ph ) ) ) <-> ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) ) )
9		19.8a 1482	. . . . 5 |- ( ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
10	9	ex 108	. . . 4 |- ( x = y -> ( ph -> E. x ( x = y /\ ph ) ) )
11		vex 2560	. . . . . 6 |- y e. _V
12	11	alexeq 2670	. . . . 5 |- ( A. x ( x = y -> ph ) <-> E. x ( x = y /\ ph ) )
13		sp 1401	. . . . . 6 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
14	13	com12 27	. . . . 5 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
15	12, 14	syl5bir 142	. . . 4 |- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
16	10, 15	impbid 120	. . 3 |- ( x = y -> ( ph <-> E. x ( x = y /\ ph ) ) )
17	8, 16	vtoclg 2613	. 2 |- ( A e. _V -> ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) )
18	3, 17	mpcom 32	1 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  ceqsexg  2672  sbc6g  2788