Theorem moi2 2930

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 2929	. . . 4 |- ( ( A e. B /\ E* x ph /\ ph ) -> ( x = A <-> ps ) )
3	2	3expa 1204	. . 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 1363   E*wmo 2037   e. wcel 2158

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751

This theorem is referenced by:  fsum3  11409  fprodseq  11605  txcn  14128