Theorem pm3.14 703

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 833. The converse holds for decidable propositions, as seen at pm3.13dc 901. (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 107	. . 3 |- ( ( ph /\ ps ) -> ph )
2	1	con3i 595	. 2 |- ( -. ph -> -. ( ph /\ ps ) )
3		simpr 108	. . 3 |- ( ( ph /\ ps ) -> ps )
4	3	con3i 595	. 2 |- ( -. ps -> -. ( ph /\ ps ) )
5	2, 4	jaoi 669	1 |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 662

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  pm3.1  704  xoranor  1309  difindiss  3234