This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem bj-19.41al

Description: Special case of 19.41 proved from core axioms, ax-10 (modal5), and hba1 (modal4). (Contributed by BJ, 29-Dec-2020) (Proof modification is discouraged.)

Ref Expression
Assertion bj-19.41al 
|- ( E. x ( ph /\ A. x ps ) <-> ( E. x ph /\ A. x ps ) )

Proof

Step Hyp Ref Expression
1 19.40 
 |-  ( E. x ( ph /\ A. x ps ) -> ( E. x ph /\ E. x A. x ps ) )
2 hbe1a 
 |-  ( E. x A. x ps -> A. x ps )
3 2 anim2i 
 |-  ( ( E. x ph /\ E. x A. x ps ) -> ( E. x ph /\ A. x ps ) )
4 1 3 syl 
 |-  ( E. x ( ph /\ A. x ps ) -> ( E. x ph /\ A. x ps ) )
5 hba1 
 |-  ( A. x ps -> A. x A. x ps )
6 5 anim2i 
 |-  ( ( E. x ph /\ A. x ps ) -> ( E. x ph /\ A. x A. x ps ) )
7 19.29r 
 |-  ( ( E. x ph /\ A. x A. x ps ) -> E. x ( ph /\ A. x ps ) )
8 6 7 syl 
 |-  ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ A. x ps ) )
9 4 8 impbii 
 |-  ( E. x ( ph /\ A. x ps ) <-> ( E. x ph /\ A. x ps ) )