Theorem moi 2913

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 2912	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )
4	3	biimprd 157	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( ch -> A = B ) )
5	4	3expia 1200	. . 3 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph ) -> ( ps -> ( ch -> A = B ) ) )
6	5	impd 252	. 2 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph ) -> ( ( ps /\ ch ) -> A = B ) )
7	6	3impia 1195	1 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   E*wmo 2020   e. wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by: (None)