Theorem reupick3 3402

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)

Assertion

Ref	Expression
reupick3	|- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem reupick3

Step	Hyp	Ref	Expression
1		df-reu 2449	. . . 4 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		df-rex 2448	. . . . 5 |- ( E. x e. A ( ph /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
3		anass 399	. . . . . 6 |- ( ( ( x e. A /\ ph ) /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
4	3	exbii 1592	. . . . 5 |- ( E. x ( ( x e. A /\ ph ) /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
5	2, 4	bitr4i 186	. . . 4 |- ( E. x e. A ( ph /\ ps ) <-> E. x ( ( x e. A /\ ph ) /\ ps ) )
6		eupick 2092	. . . 4 |- ( ( E! x ( x e. A /\ ph ) /\ E. x ( ( x e. A /\ ph ) /\ ps ) ) -> ( ( x e. A /\ ph ) -> ps ) )
7	1, 5, 6	syl2anb 289	. . 3 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) ) -> ( ( x e. A /\ ph ) -> ps ) )
8	7	expd 256	. 2 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) ) -> ( x e. A -> ( ph -> ps ) ) )
9	8	3impia 1189	1 |- ( ( E! x e. A ph /\ E. x e. A ( ph /\ ps ) /\ x e. A ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 967   E.wex 1479   E!weu 2013   e. wcel 2135   E.wrex 2443   E!wreu 2444

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522

This theorem depends on definitions:  df-bi 116  df-3an 969  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-rex 2448  df-reu 2449

This theorem is referenced by:  reupick2  3403