Theorem bj-19.41al 32637

Description: Special case of 19.41 2103 proved from Tarski, ax-10 2019 (modal5) and hba1 2151 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem bj-19.41al

Step	Hyp	Ref	Expression
1		19.40 1797	. . 3 |- ( E. x ( ph /\ A. x ps ) -> ( E. x ph /\ E. x A. x ps ) )
2		bj-modal5e 32636	. . . 4 |- ( E. x A. x ps -> A. x ps )
3	2	anim2i 593	. . 3 |- ( ( E. x ph /\ E. x A. x ps ) -> ( E. x ph /\ A. x ps ) )
4	1, 3	syl 17	. 2 |- ( E. x ( ph /\ A. x ps ) -> ( E. x ph /\ A. x ps ) )
5		hba1 2151	. . . 4 |- ( A. x ps -> A. x A. x ps )
6	5	anim2i 593	. . 3 |- ( ( E. x ph /\ A. x ps ) -> ( E. x ph /\ A. x A. x ps ) )
7		19.29r 1802	. . 3 |- ( ( E. x ph /\ A. x A. x ps ) -> E. x ( ph /\ A. x ps ) )
8	6, 7	syl 17	. 2 |- ( ( E. x ph /\ A. x ps ) -> E. x ( ph /\ A. x ps ) )
9	4, 8	impbii 199	1 |- ( E. x ( ph /\ A. x ps ) <-> ( E. x ph /\ A. x ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bj-equsexval  32638