Theorem pm3.14 761

Description: Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 907. The converse holds for decidable propositions, as seen at pm3.13dc 968. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

Ref	Expression
pm3.14	|- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )

Proof of Theorem pm3.14

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( ph /\ ps ) -> ph )
2	1	con3i 637	. 2 |- ( -. ph -> -. ( ph /\ ps ) )
3		simpr 110	. . 3 |- ( ( ph /\ ps ) -> ps )
4	3	con3i 637	. 2 |- ( -. ps -> -. ( ph /\ ps ) )
5	2, 4	jaoi 724	1 |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716

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-in1 619  ax-in2 620  ax-io 717

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm3.1  762  xoranor  1422  difindiss  3463  swrdnd  11306