Metamath Proof Explorer


Theorem ralrexbidOLD

Description: Obsolete version of ralrexbid as of 4-Nov-2024. (Contributed by AV, 21-Oct-2023) Reduce axiom usage. (Revised by SN, 13-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis ralrexbid.1
|- ( ph -> ( ps <-> th ) )
Assertion ralrexbidOLD
|- ( A. x e. A ph -> ( E. x e. A ps <-> E. x e. A th ) )

Proof

Step Hyp Ref Expression
1 ralrexbid.1
 |-  ( ph -> ( ps <-> th ) )
2 df-ral
 |-  ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3 1 imim2i
 |-  ( ( x e. A -> ph ) -> ( x e. A -> ( ps <-> th ) ) )
4 3 pm5.32d
 |-  ( ( x e. A -> ph ) -> ( ( x e. A /\ ps ) <-> ( x e. A /\ th ) ) )
5 4 alexbii
 |-  ( A. x ( x e. A -> ph ) -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. A /\ th ) ) )
6 2 5 sylbi
 |-  ( A. x e. A ph -> ( E. x ( x e. A /\ ps ) <-> E. x ( x e. A /\ th ) ) )
7 df-rex
 |-  ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
8 df-rex
 |-  ( E. x e. A th <-> E. x ( x e. A /\ th ) )
9 6 7 8 3bitr4g
 |-  ( A. x e. A ph -> ( E. x e. A ps <-> E. x e. A th ) )