Theorem eqreu 2871

Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypothesis

Ref	Expression
eqreu.1	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
eqreu	|- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) -> E! x e. A ph )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem eqreu

Step	Hyp	Ref	Expression
1		ralbiim 2564	. . . . 5 |- ( A. x e. A ( ph <-> x = B ) <-> ( A. x e. A ( ph -> x = B ) /\ A. x e. A ( x = B -> ph ) ) )
2		eqreu.1	. . . . . . 7 |- ( x = B -> ( ph <-> ps ) )
3	2	ceqsralv 2712	. . . . . 6 |- ( B e. A -> ( A. x e. A ( x = B -> ph ) <-> ps ) )
4	3	anbi2d 459	. . . . 5 |- ( B e. A -> ( ( A. x e. A ( ph -> x = B ) /\ A. x e. A ( x = B -> ph ) ) <-> ( A. x e. A ( ph -> x = B ) /\ ps ) ) )
5	1, 4	syl5bb 191	. . . 4 |- ( B e. A -> ( A. x e. A ( ph <-> x = B ) <-> ( A. x e. A ( ph -> x = B ) /\ ps ) ) )
6		reu6i 2870	. . . . 5 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )
7	6	ex 114	. . . 4 |- ( B e. A -> ( A. x e. A ( ph <-> x = B ) -> E! x e. A ph ) )
8	5, 7	sylbird 169	. . 3 |- ( B e. A -> ( ( A. x e. A ( ph -> x = B ) /\ ps ) -> E! x e. A ph ) )
9	8	3impib 1179	. 2 |- ( ( B e. A /\ A. x e. A ( ph -> x = B ) /\ ps ) -> E! x e. A ph )
10	9	3com23 1187	1 |- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2414   E!wreu 2416

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 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-v 2683

This theorem is referenced by: (None)