Theorem 19.31r 1692

Description: One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)

Hypothesis

Ref	Expression
19.31r.1	|- F/ x ps

Assertion

Ref	Expression
19.31r	|- ( ( A. x ph \/ ps ) -> A. x ( ph \/ ps ) )

Proof of Theorem 19.31r

Step	Hyp	Ref	Expression
1		19.31r.1	. . 3 |- F/ x ps
2	1	19.32r 1691	. 2 |- ( ( ps \/ A. x ph ) -> A. x ( ps \/ ph ) )
3		orcom 729	. 2 |- ( ( A. x ph \/ ps ) <-> ( ps \/ A. x ph ) )
4		orcom 729	. . 3 |- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
5	4	albii 1481	. 2 |- ( A. x ( ph \/ ps ) <-> A. x ( ps \/ ph ) )
6	2, 3, 5	3imtr4i 201	1 |- ( ( A. x ph \/ ps ) -> A. x ( ph \/ ps ) )

Syntax hints:   -> wi 4   \/ wo 709   A.wal 1362   F/wnf 1471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-gen 1460  ax-4 1521

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by: (None)