Theorem moi 3389

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 3388	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( A = B <-> ch ) )
4	3	biimprd 238	. . . 4 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ps ) -> ( ch -> A = B ) )
5	4	3expia 1267	. . 3 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph ) -> ( ps -> ( ch -> A = B ) ) )
6	5	impd 447	. 2 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph ) -> ( ( ps /\ ch ) -> A = B ) )
7	6	3impia 1261	1 |- ( ( ( A e. C /\ B e. D ) /\ E* x ph /\ ( ps /\ ch ) ) -> A = B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  enqeq  9756  f1otrspeq  17867  hausflim  21785  tglineineq  25538  tglineinteq  25540