Theorem 19.32r 1570

Description: One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if ph is decidable, as seen at 19.32dc 1569. (Contributed by Jim Kingdon, 28-Jul-2018.)

Hypothesis

Ref	Expression
19.32r.1	|- F/ x ph

Assertion

Ref	Expression
19.32r	|- ( ( ph \/ A. x ps ) -> A. x ( ph \/ ps ) )

Proof of Theorem 19.32r

Step	Hyp	Ref	Expression
1		19.32r.1	. . 3 |- F/ x ph
2		orc 633	. . 3 |- ( ph -> ( ph \/ ps ) )
3	1, 2	alrimi 1415	. 2 |- ( ph -> A. x ( ph \/ ps ) )
4		olc 632	. . 3 |- ( ps -> ( ph \/ ps ) )
5	4	alimi 1344	. 2 |- ( A. x ps -> A. x ( ph \/ ps ) )
6	3, 5	jaoi 636	1 |- ( ( ph \/ A. x ps ) -> A. x ( ph \/ ps ) )

Syntax hints:   -> wi 4   \/ wo 629   A.wal 1241   F/wnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-gen 1338  ax-4 1400

This theorem depends on definitions:  df-bi 110  df-nf 1350

This theorem is referenced by:  19.31r  1571