Theorem 19.40b 1815

Description: The antecedent provides a condition implying the converse of 19.40 1797. This is to 19.40 1797 what 19.33b 1813 is to 19.33 1812. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)

Assertion

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

Proof of Theorem 19.40b

Step	Hyp	Ref	Expression
1		pm3.21 464	. . . . 5 |- ( ps -> ( ph -> ( ph /\ ps ) ) )
2	1	aleximi 1759	. . . 4 |- ( A. x ps -> ( E. x ph -> E. x ( ph /\ ps ) ) )
3		pm3.2 463	. . . . 5 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
4	3	aleximi 1759	. . . 4 |- ( A. x ph -> ( E. x ps -> E. x ( ph /\ ps ) ) )
5	2, 4	jaoa 532	. . 3 |- ( ( A. x ps \/ A. x ph ) -> ( ( E. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) )
6	5	orcoms 404	. 2 |- ( ( A. x ph \/ A. x ps ) -> ( ( E. x ph /\ E. x ps ) -> E. x ( ph /\ ps ) ) )
7		19.40 1797	. 2 |- ( E. x ( ph /\ ps ) -> ( E. x ph /\ E. x ps ) )
8	6, 7	impbid1 215	1 |- ( ( A. x ph \/ A. x ps ) -> ( ( E. x ph /\ E. x ps ) <-> E. x ( ph /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ 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

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

This theorem is referenced by: (None)