Theorem moi2 3007

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 3006	. . . 4 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )
3	2	3expa 1234	. . 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 1402   E*wmo 2087   e. wcel 2209

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-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  fsum3  12132  fprodseq  12328  txcn  15299