Theorem pm3.14 760

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 906. The converse holds for decidable propositions, as seen at pm3.13dc 967. (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 723	1 |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715

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 716

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm3.1  761  xoranor  1421  difindiss  3461  swrdnd  11239