Theorem moi2 2958

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 2957	. . . 4 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )
3	2	3expa 1206	. . 3 |- ( ( ( A e. B /\ E* x ph ) /\ ph ) -> ( x = A <-> ps ) )
4	3	biimprd 158	. 2 |- ( ( ( A e. B /\ E* x ph ) /\ ph ) -> ( ps -> x = A ) )
5	4	impr 379	1 |- ( ( ( A e. B /\ E* x ph ) /\ ( ph /\ ps ) ) -> x = A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E*wmo 2056   e. wcel 2177

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775

This theorem is referenced by:  fsum3  11773  fprodseq  11969  txcn  14822