Theorem pm3.14 669

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 798. The converse holds for decidable propositions, as seen at pm3.13dc 865. (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 102	. . 3 |- ((ph /\ ps) -> ph)
2	1	con3i 561	. 2 |- (-. ph -> -. (ph /\ ps))
3		simpr 103	. . 3 |- ((ph /\ ps) -> ps)
4	3	con3i 561	. 2 |- (-. ps -> -. (ph /\ ps))
5	2, 4	jaoi 635	1 |- ((-. ph \/ -. ps) -> -. (ph /\ ps))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   \/ wo 628

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  pm3.1  670  xoranor  1267  difindiss  3185