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Metamath Proof Explorer

Theorem reuxfr1dd

Description: Transfer existential uniqueness from a variable x to another variable y contained in expression A . Simplifies reuxfr1d . (Contributed by Zhi Wang, 20-Sep-2025)

Ref Expression
Hypotheses reuxfr1dd.1 
|- ( ( ph /\ y e. C ) -> A e. B )
reuxfr1dd.2 
|- ( ( ph /\ x e. B ) -> E! y e. C x = A )
reuxfr1dd.3 
|- ( ( ph /\ ( y e. C /\ x = A ) ) -> ( ps <-> ch ) )
Assertion reuxfr1dd 
|- ( ph -> ( E! x e. B ps <-> E! y e. C ch ) )

Proof

Step Hyp Ref Expression
1 reuxfr1dd.1 
 |-  ( ( ph /\ y e. C ) -> A e. B )
2 reuxfr1dd.2 
 |-  ( ( ph /\ x e. B ) -> E! y e. C x = A )
3 reuxfr1dd.3 
 |-  ( ( ph /\ ( y e. C /\ 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 -> ( ( ( y e. C /\ x = A ) /\ ps ) <-> ( ( y e. C /\ x = A ) /\ ch ) ) )
9 anass 
 |-  ( ( ( y e. C /\ x = A ) /\ ps ) <-> ( y e. C /\ ( x = A /\ ps ) ) )
10 anass 
 |-  ( ( ( y e. C /\ x = A ) /\ ch ) <-> ( y e. C /\ ( x = A /\ ch ) ) )
11 8 9 10 3bitr3g 
 |-  ( ph -> ( ( y e. C /\ ( x = A /\ ps ) ) <-> ( y e. C /\ ( x = A /\ ch ) ) ) )
12 11 rexbidv2 
 |-  ( ph -> ( E. y e. C ( x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
13 7 12 bitr3id 
 |-  ( ph -> ( ( E. y e. C x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
14 13 adantr 
 |-  ( ( ph /\ x e. B ) -> ( ( E. y e. C x = A /\ ps ) <-> E. y e. C ( x = A /\ ch ) ) )
15 6 14 bitrd 
 |-  ( ( ph /\ x e. B ) -> ( ps <-> E. y e. C ( x = A /\ ch ) ) )
16 15 reubidva 
 |-  ( ph -> ( E! x e. B ps <-> E! x e. B E. y e. C ( x = A /\ ch ) ) )
17 reurmo 
 |-  ( E! y e. C x = A -> E* y e. C x = A )
18 2 17 syl 
 |-  ( ( ph /\ x e. B ) -> E* y e. C x = A )
19 1 18 reuxfrd 
 |-  ( ph -> ( E! x e. B E. y e. C ( x = A /\ ch ) <-> E! y e. C ch ) )
20 16 19 bitrd 
 |-  ( ph -> ( E! x e. B ps <-> E! y e. C ch ) )