Metamath Proof Explorer


Theorem ralprgOLD

Description: Obsolete version of ralprg as of 30-Sep-2024. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015) (Proof shortened by AV, 8-Apr-2023) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ralprg.1
|- ( x = A -> ( ph <-> ps ) )
ralprg.2
|- ( x = B -> ( ph <-> ch ) )
Assertion ralprgOLD
|- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )

Proof

Step Hyp Ref Expression
1 ralprg.1
 |-  ( x = A -> ( ph <-> ps ) )
2 ralprg.2
 |-  ( x = B -> ( ph <-> ch ) )
3 nfv
 |-  F/ x ps
4 nfv
 |-  F/ x ch
5 3 4 1 2 ralprgf
 |-  ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )