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Metamath Proof Explorer

Theorem 19.40b

Description: The antecedent provides a condition implying the converse of 19.40 . This is to 19.40 what 19.33b is to 19.33 . (Contributed by BJ, 6-May-2019) (Proof shortened by Wolf Lammen, 13-Nov-2020)

Ref Expression
Assertion 19.40b 
|- ( ( A. x ph \/ A. x ps ) -> ( ( E. x ph /\ E. x ps ) <-> E. x ( ph /\ ps ) ) )

Proof

Step Hyp Ref Expression
1 pm3.21 
 |-  ( ps -> ( ph -> ( ph /\ ps ) ) )
2 1 aleximi 
 |-  ( A. x ps -> ( E. x ph -> E. x ( ph /\ ps ) ) )
3 pm3.2 
 |-  ( ph -> ( ps -> ( ph /\ ps ) ) )
4 3 aleximi 
 |-  ( A. x ph -> ( E. x ps -> E. x ( ph /\ ps ) ) )
5 2 4 jaoa 
 |-  ( ( A. x ps \/ A. x ph ) -> ( ( E. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) )
6 5 orcoms 
 |-  ( ( A. x ph \/ A. x ps ) -> ( ( E. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) )
7 19.40 
 |-  ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
8 6 7 impbid1 
 |-  ( ( A. x ph \/ A. x ps ) -> ( ( E. x ph /\ E. x ps ) <-> E. x ( ph /\ ps ) ) )