Theorem ceqex 2862

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 1588	. . 3 |- ( x = A -> E. x x = A )
2		isset 2741	. . 3 |- ( A e. _V <-> E. x x = A )
3	1, 2	sylibr 134	. 2 |- ( x = A -> A e. _V )
4		eqeq2 2185	. . . 4 |- ( y = A -> ( x = y <-> x = A ) )
5	4	anbi1d 465	. . . . . 6 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
6	5	exbidv 1823	. . . . 5 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
7	6	bibi2d 232	. . . 4 |- ( y = A -> ( ( ph <-> E. x ( x = y /\ ph ) ) <-> ( ph <-> E. x ( x = A /\ ph ) ) ) )
8	4, 7	imbi12d 234	. . 3 |- ( y = A -> ( ( x = y -> ( ph <-> E. x ( x = y /\ ph ) ) ) <-> ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) ) )
9		19.8a 1588	. . . . 5 |- ( ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
10	9	ex 115	. . . 4 |- ( x = y -> ( ph -> E. x ( x = y /\ ph ) ) )
11		vex 2738	. . . . . 6 |- y e. _V
12	11	alexeq 2861	. . . . 5 |- ( A. x ( x = y -> ph ) <-> E. x ( x = y /\ ph ) )
13		sp 1509	. . . . . 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 153	. . . 4 |- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
16	10, 15	impbid 129	. . 3 |- ( x = y -> ( ph <-> E. x ( x = y /\ ph ) ) )
17	8, 16	vtoclg 2795	. 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 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1490   e. wcel 2146   _Vcvv 2735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737

This theorem is referenced by:  ceqsexg  2863  sbc6g  2985