Theorem 19.31r 1669

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 1668	. 2 |- ( ( ps \/ A. x ph ) -> A. x ( ps \/ ph ) )
3		orcom 718	. 2 |- ( ( A. x ph \/ ps ) <-> ( ps \/ A. x ph ) )
4		orcom 718	. . 3 |- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
5	4	albii 1458	. 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 698   A.wal 1341   F/wnf 1448

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 699  ax-5 1435  ax-gen 1437  ax-4 1498

This theorem depends on definitions:  df-bi 116  df-nf 1449

This theorem is referenced by: (None)