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Metamath Proof Explorer

Theorem mopickr

Description: "At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 ( pm2.21 ) and *14.26 ( eupickbi ) from WhiteheadRussell p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024) (Proof modification is discouraged.)

Ref Expression
Assertion mopickr 
|- ( ( E* x ps /\ E. x ( ph /\ ps ) ) -> ( ps -> ph ) )

Proof

Step Hyp Ref Expression
1 exancom 
 |-  ( E. x ( ph /\ ps ) <-> E. x ( ps /\ ph ) )
2 moeu2 
 |-  ( E* x ps <-> ( -. E. x ps \/ E! x ps ) )
3 19.8a 
 |-  ( ps -> E. x ps )
4 3 con3i 
 |-  ( -. E. x ps -> -. ps )
5 pm2.21 
 |-  ( -. ps -> ( ps -> ph ) )
6 4 5 syl 
 |-  ( -. E. x ps -> ( ps -> ph ) )
7 6 a1d 
 |-  ( -. E. x ps -> ( E. x ( ps /\ ph ) -> ( ps -> ph ) ) )
8 eupickbi 
 |-  ( E! x ps -> ( E. x ( ps /\ ph ) <-> A. x ( ps -> ph ) ) )
9 sp 
 |-  ( A. x ( ps -> ph ) -> ( ps -> ph ) )
10 8 9 biimtrdi 
 |-  ( E! x ps -> ( E. x ( ps /\ ph ) -> ( ps -> ph ) ) )
11 7 10 jaoi 
 |-  ( ( -. E. x ps \/ E! x ps ) -> ( E. x ( ps /\ ph ) -> ( ps -> ph ) ) )
12 2 11 sylbi 
 |-  ( E* x ps -> ( E. x ( ps /\ ph ) -> ( ps -> ph ) ) )
13 1 12 biimtrid 
 |-  ( E* x ps -> ( E. x ( ph /\ ps ) -> ( ps -> ph ) ) )
14 13 imp 
 |-  ( ( E* x ps /\ E. x ( ph /\ ps ) ) -> ( ps -> ph ) )