Metamath Proof Explorer

Theorem rmoeq1

Description: Equality theorem for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017) Remove usage of ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 30-Apr-2023) Avoid ax-8 . (Revised by Wolf Lammen, 12-Mar-2025)

Ref Expression
Assertion rmoeq1 
|- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )

Proof

Step Hyp Ref Expression
1 dfcleq 
 |-  ( A = B <-> A. x ( x e. A <-> x e. B ) )
2 1 biimpi 
 |-  ( A = B -> A. x ( x e. A <-> x e. B ) )
3 anbi1 
 |-  ( ( x e. A <-> x e. B ) -> ( ( x e. A /\ ph ) <-> ( x e. B /\ ph ) ) )
4 3 imbi1d 
 |-  ( ( x e. A <-> x e. B ) -> ( ( ( x e. A /\ ph ) -> x = z ) <-> ( ( x e. B /\ ph ) -> x = z ) ) )
5 4 alimi 
 |-  ( A. x ( x e. A <-> x e. B ) -> A. x ( ( ( x e. A /\ ph ) -> x = z ) <-> ( ( x e. B /\ ph ) -> x = z ) ) )
6 albi 
 |-  ( A. x ( ( ( x e. A /\ ph ) -> x = z ) <-> ( ( x e. B /\ ph ) -> x = z ) ) -> ( A. x ( ( x e. A /\ ph ) -> x = z ) <-> A. x ( ( x e. B /\ ph ) -> x = z ) ) )
7 2 5 6 3syl 
 |-  ( A = B -> ( A. x ( ( x e. A /\ ph ) -> x = z ) <-> A. x ( ( x e. B /\ ph ) -> x = z ) ) )
8 7 exbidv 
 |-  ( A = B -> ( E. z A. x ( ( x e. A /\ ph ) -> x = z ) <-> E. z A. x ( ( x e. B /\ ph ) -> x = z ) ) )
9 df-mo 
 |-  ( E* x ( x e. A /\ ph ) <-> E. z A. x ( ( x e. A /\ ph ) -> x = z ) )
10 df-mo 
 |-  ( E* x ( x e. B /\ ph ) <-> E. z A. x ( ( x e. B /\ ph ) -> x = z ) )
11 8 9 10 3bitr4g 
 |-  ( A = B -> ( E* x ( x e. A /\ ph ) <-> E* x ( x e. B /\ ph ) ) )
12 df-rmo 
 |-  ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
13 df-rmo 
 |-  ( E* x e. B ph <-> E* x ( x e. B /\ ph ) )
14 11 12 13 3bitr4g 
 |-  ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )