Theorem ceqex 2722

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 1522	. . 3 |- ( x = A -> E. x x = A )
2		isset 2605	. . 3 |- ( A e. _V <-> E. x x = A )
3	1, 2	sylibr 132	. 2 |- ( x = A -> A e. _V )
4		eqeq2 2090	. . . 4 |- ( y = A -> ( x = y <-> x = A ) )
5	4	anbi1d 452	. . . . . 6 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
6	5	exbidv 1746	. . . . 5 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
7	6	bibi2d 230	. . . 4 |- ( y = A -> ( ( ph <-> E. x ( x = y /\ ph ) ) <-> ( ph <-> E. x ( x = A /\ ph ) ) ) )
8	4, 7	imbi12d 232	. . 3 |- ( y = A -> ( ( x = y -> ( ph <-> E. x ( x = y /\ ph ) ) ) <-> ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) ) )
9		19.8a 1522	. . . . 5 |- ( ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
10	9	ex 113	. . . 4 |- ( x = y -> ( ph -> E. x ( x = y /\ ph ) ) )
11		vex 2604	. . . . . 6 |- y e. _V
12	11	alexeq 2721	. . . . 5 |- ( A. x ( x = y -> ph ) <-> E. x ( x = y /\ ph ) )
13		sp 1441	. . . . . 6 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
14	13	com12 30	. . . . 5 |- ( x = y -> ( A. x ( x = y -> ph ) -> ph ) )
15	12, 14	syl5bir 151	. . . 4 |- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
16	10, 15	impbid 127	. . 3 |- ( x = y -> ( ph <-> E. x ( x = y /\ ph ) ) )
17	8, 16	vtoclg 2658	. 2 |- ( A e. _V -> ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) )
18	3, 17	mpcom 36	1 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  ceqsexg  2723  sbc6g  2839