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Metamath Proof Explorer

Theorem reuxfr1d

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . Cf. reuxfr1ds . (Contributed by Thierry Arnoux, 7-Apr-2017)

Ref Expression
Hypotheses reuxfr1d.1 
|- ( ( ph /\ y e. C ) -> A e. B )
reuxfr1d.2 
|- ( ( ph /\ x e. B ) -> E! y e. C x = A )
reuxfr1d.3 
|- ( ( ph /\ x = A ) -> ( ps <-> ch ) )
Assertion reuxfr1d 
|- ( ph -> ( E! x e. B ps <-> E! y e. C ch ) )

Proof

Step Hyp Ref Expression
1 reuxfr1d.1 
 |-  ( ( ph /\ y e. C ) -> A e. B )
2 reuxfr1d.2 
 |-  ( ( ph /\ x e. B ) -> E! y e. C x = A )
3 reuxfr1d.3 
 |-  ( ( ph /\ x = A ) -> ( ps <-> ch ) )
4 reurex 
 |-  ( E! y e. C x = A -> E. y e. C x = A )
5 2 4 syl 
 |-  ( ( ph /\ x e. B ) -> E. y e. C x = A )
6 5 biantrurd 
 |-  ( ( ph /\ x e. B ) -> ( ps <-> ( E. y e. C x = A /\ ps ) ) )
7 r19.41v 
 |-  ( E. y e. C ( x = A /\ ps ) <-> ( E. y e. C x = A /\ ps ) )
8 3 pm5.32da 
 |-  ( ph -> ( ( x = A /\ ps ) <-> ( x = A /\ ch ) ) )
9 8 rexbidv 
 |-  ( ph -> ( E. y e. C ( x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
10 7 9 bitr3id 
 |-  ( ph -> ( ( E. y e. C x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
11 10 adantr 
 |-  ( ( ph /\ x e. B ) -> ( ( E. y e. C x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
12 6 11 bitrd 
 |-  ( ( ph /\ x e. B ) -> ( ps <-> E. y e. C ( x = A /\ ch ) ) )
13 12 reubidva 
 |-  ( ph -> ( E! x e. B ps <-> E! x e. B E. y e. C ( x = A /\ ch ) ) )
14 reurmo 
 |-  ( E! y e. C x = A -> E* y e. C x = A )
15 2 14 syl 
 |-  ( ( ph /\ x e. B ) -> E* y e. C x = A )
16 1 15 reuxfrd 
 |-  ( ph -> ( E! x e. B E. y e. C ( x = A /\ ch ) <-> E! y e. C ch ) )
17 13 16 bitrd 
 |-  ( ph -> ( E! x e. B ps <-> E! y e. C ch ) )