Theorem moi2 3387

Description: Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)

Hypothesis

Ref	Expression
moi2.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
moi2	|- ( ( ( A e. B /\ E* x ph ) /\ ( ph /\ ps ) ) -> x = A )

Distinct variable groups:   x, A   ps, x

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

Proof of Theorem moi2

Step	Hyp	Ref	Expression
1		moi2.1	. . . . 5 |- ( x = A -> ( ph <-> ps ) )
2	1	mob2 3386	. . . 4 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )
3	2	3expa 1265	. . 3 |- ( ( ( A e. B /\ E* x ph ) /\ ph ) -> ( x = A <-> ps ) )
4	3	biimprd 238	. 2 |- ( ( ( A e. B /\ E* x ph ) /\ ph ) -> ( ps -> x = A ) )
5	4	impr 649	1 |- ( ( ( A e. B /\ E* x ph ) /\ ( ph /\ ps ) ) -> x = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  fsum  14451  fprod  14671  txcn  21429  haustsms2  21940