Theorem eqreu 3018

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 2685	. . . . 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 2853	. . . . . 6 |- ( B e. A -> ( A. x e. A ( x = B -> ph ) <-> ps ) )
4	3	anbi2d 468	. . . . 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	bitrid 192	. . . 4 |- ( B e. A -> ( A. x e. A ( ph <-> x = B ) <-> ( A. x e. A ( ph -> x = B ) /\ ps ) ) )
6		reu6i 3017	. . . . 5 |- ( ( B e. A /\ A. x e. A ( ph <-> x = B ) ) -> E! x e. A ph )
7	6	ex 115	. . . 4 |- ( B e. A -> ( A. x e. A ( ph <-> x = B ) -> E! x e. A ph ) )
8	5, 7	sylbird 170	. . 3 |- ( B e. A -> ( ( A. x e. A ( ph -> x = B ) /\ ps ) -> E! x e. A ph ) )
9	8	3impib 1232	. 2 |- ( ( B e. A /\ A. x e. A ( ph -> x = B ) /\ ps ) -> E! x e. A ph )
10	9	3com23 1240	1 |- ( ( B e. A /\ ps /\ A. x e. A ( ph -> x = B ) ) -> E! x e. A ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E!wreu 2530

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823

This theorem is referenced by:  depind  16664