Theorem moi 2922

Description: Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)

Hypotheses

Ref	Expression
moi.1	|- ( x = A -> ( ph <-> ps ) )
moi.2	|- ( x = B -> ( ph <-> ch ) )

Assertion

Ref	Expression
moi	|- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) -> A = B )

Distinct variable groups:   x, A   x, B   ch, x   ps, x

Allowed substitution hints:   ph( x)   C( x)   D( x)

Proof of Theorem moi

Step	Hyp	Ref	Expression
1		moi.1	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
2		moi.2	. . . . . 6 |- ( x = B -> ( ph <-> ch ) )
3	1, 2	mob 2921	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )
4	3	biimprd 158	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( ch -> A = B ) )
5	4	3expia 1205	. . 3 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph ) -> ( ps -> ( ch -> A = B ) ) )
6	5	impd 254	. 2 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph ) -> ( ( ps /\ ch ) -> A = B ) )
7	6	3impia 1200	1 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   E*wmo 2027   e. wcel 2148

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741

This theorem is referenced by: (None)