Theorem ceqex 2812

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 1569	. . 3 |- ( x = A -> E. x x = A )
2		isset 2692	. . 3 |- ( A e. _V <-> E. x x = A )
3	1, 2	sylibr 133	. 2 |- ( x = A -> A e. _V )
4		eqeq2 2149	. . . 4 |- ( y = A -> ( x = y <-> x = A ) )
5	4	anbi1d 460	. . . . . 6 |- ( y = A -> ( ( x = y /\ ph ) <-> ( x = A /\ ph ) ) )
6	5	exbidv 1797	. . . . 5 |- ( y = A -> ( E. x ( x = y /\ ph ) <-> E. x ( x = A /\ ph ) ) )
7	6	bibi2d 231	. . . 4 |- ( y = A -> ( ( ph <-> E. x ( x = y /\ ph ) ) <-> ( ph <-> E. x ( x = A /\ ph ) ) ) )
8	4, 7	imbi12d 233	. . 3 |- ( y = A -> ( ( x = y -> ( ph <-> E. x ( x = y /\ ph ) ) ) <-> ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) ) ) )
9		19.8a 1569	. . . . 5 |- ( ( x = y /\ ph ) -> E. x ( x = y /\ ph ) )
10	9	ex 114	. . . 4 |- ( x = y -> ( ph -> E. x ( x = y /\ ph ) ) )
11		vex 2689	. . . . . 6 |- y e. _V
12	11	alexeq 2811	. . . . 5 |- ( A. x ( x = y -> ph ) <-> E. x ( x = y /\ ph ) )
13		sp 1488	. . . . . 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 152	. . . 4 |- ( x = y -> ( E. x ( x = y /\ ph ) -> ph ) )
16	10, 15	impbid 128	. . 3 |- ( x = y -> ( ph <-> E. x ( x = y /\ ph ) ) )
17	8, 16	vtoclg 2746	. 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 103   <-> wb 104   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  ceqsexg  2813  sbc6g  2933