Theorem reupick3 3541

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 2622	. . . 4 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
2		df-rex 2621	. . . . 5 |- (E.x e. A (ph /\ ps) <-> E.x(x e. A /\ (ph /\ ps)))
3		anass 630	. . . . . 6 |- (((x e. A /\ ph) /\ ps) <-> (x e. A /\ (ph /\ ps)))
4	3	exbii 1582	. . . . 5 |- (E.x((x e. A /\ ph) /\ ps) <-> E.x(x e. A /\ (ph /\ ps)))
5	2, 4	bitr4i 243	. . . 4 |- (E.x e. A (ph /\ ps) <-> E.x((x e. A /\ ph) /\ ps))
6		eupick 2267	. . . 4 |- ((E!x(x e. A /\ ph) /\ E.x((x e. A /\ ph) /\ ps)) -> ((x e. A /\ ph) -> ps))
7	1, 5, 6	syl2anb 465	. . 3 |- ((E!x e. A ph /\ E.x e. A (ph /\ ps)) -> ((x e. A /\ ph) -> ps))
8	7	exp3a 425	. 2 |- ((E!x e. A ph /\ E.x e. A (ph /\ ps)) -> (x e. A -> (ph -> ps)))
9	8	3impia 1148	1 |- ((E!x e. A ph /\ E.x e. A (ph /\ ps) /\ x e. A) -> (ph -> ps))

Syntax hints:   -> wi 4   /\ wa 358   /\ w3a 934  E.wex 1541   e. wcel 1710  E!weu 2204  E.wrex 2616  E!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-rex 2621  df-reu 2622

This theorem is referenced by:  reupick2  3542