Theorem pm3.14 758

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 904. The converse holds for decidable propositions, as seen at pm3.13dc 965. (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 635	. 2 |- ( -. ph -> -. ( ph /\ ps ) )
3		simpr 110	. . 3 |- ( ( ph /\ ps ) -> ps )
4	3	con3i 635	. 2 |- ( -. ps -> -. ( ph /\ ps ) )
5	2, 4	jaoi 721	1 |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm3.1  759  xoranor  1419  difindiss  3458  swrdnd  11186