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Metamath Proof Explorer

Theorem reusv3i

Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012)

Ref Expression
Hypotheses reusv3.1 
|- ( y = z -> ( ph <-> ps ) )
reusv3.2 
|- ( y = z -> C = D )
Assertion reusv3i 
|- ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )

Proof

Step Hyp Ref Expression
1 reusv3.1 
 |-  ( y = z -> ( ph <-> ps ) )
2 reusv3.2 
 |-  ( y = z -> C = D )
3 2 eqeq2d 
 |-  ( y = z -> ( x = C <-> x = D ) )
4 1 3 imbi12d 
 |-  ( y = z -> ( ( ph -> x = C ) <-> ( ps -> x = D ) ) )
5 4 cbvralvw 
 |-  ( A. y e. B ( ph -> x = C ) <-> A. z e. B ( ps -> x = D ) )
6 5 biimpi 
 |-  ( A. y e. B ( ph -> x = C ) -> A. z e. B ( ps -> x = D ) )
7 raaanv 
 |-  ( A. y e. B A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) <-> ( A. y e. B ( ph -> x = C ) /\ A. z e. B ( ps -> x = D ) ) )
8 anim12 
 |-  ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> ( x = C /\ x = D ) ) )
9 eqtr2 
 |-  ( ( x = C /\ x = D ) -> C = D )
10 8 9 syl6 
 |-  ( ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> ( ( ph /\ ps ) -> C = D ) )
11 10 2ralimi 
 |-  ( A. y e. B A. z e. B ( ( ph -> x = C ) /\ ( ps -> x = D ) ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
12 7 11 sylbir 
 |-  ( ( A. y e. B ( ph -> x = C ) /\ A. z e. B ( ps -> x = D ) ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
13 6 12 mpdan 
 |-  ( A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )
14 13 rexlimivw 
 |-  ( E. x e. A A. y e. B ( ph -> x = C ) -> A. y e. B A. z e. B ( ( ph /\ ps ) -> C = D ) )