Theorem 19.31r 1674

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 1673	. 2 |- ( ( ps \/ A. x ph ) -> A. x ( ps \/ ph ) )
3		orcom 723	. 2 |- ( ( A. x ph \/ ps ) <-> ( ps \/ A. x ph ) )
4		orcom 723	. . 3 |- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
5	4	albii 1463	. 2 |- ( A. x ( ph \/ ps ) <-> A. x ( ps \/ ph ) )
6	2, 3, 5	3imtr4i 200	1 |- ( ( A. x ph \/ ps ) -> A. x ( ph \/ ps ) )

Syntax hints:   -> wi 4   \/ wo 703   A.wal 1346   F/wnf 1453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-gen 1442  ax-4 1503

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by: (None)