Theorem ceqex 2742

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 1527	. . 3 |- ( x = A -> E. x x = A )
2		isset 2625	. . 3 |- ( A e. _V <-> E. x x = A )
3	1, 2	sylibr 132	. 2 |- ( x = A -> A e. _V )
4		eqeq2 2097	. . . 4 |- ( y = A -> ( x = y <-> x = A ) )
5	4	anbi1d 453	. . . . . 6 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
6	5	exbidv 1753	. . . . 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 1527	. . . . 5 |- ( ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
10	9	ex 113	. . . 4 |- ( x = y -> ( ph -> E. x ( x = y /\ ph ) ) )
11		vex 2622	. . . . . 6 |- y e. _V
12	11	alexeq 2741	. . . . 5 |- ( A. x ( x = y -> ph ) <-> E. x ( x = y /\ ph ) )
13		sp 1446	. . . . . 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 2679	. 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 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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621

This theorem is referenced by:  ceqsexg  2743  sbc6g  2862