Theorem bicon2 403

Description: Contraposition. Theorem *4.12 of [WhiteheadRussell] p. 117.

Assertion

Ref	Expression
bicon2	|- ((ph <-> -. ps) <-> (ps <-> -. ph))

Proof of Theorem bicon2

Step	Hyp	Ref	Expression
1		bi2.03 144	. . 3 |- ((ph -> -. ps) <-> (ps -> -. ph))
2		bi2.15 145	. . 3 |- ((-. ps -> ph) <-> (-. ph -> ps))
3	1, 2	anbi12i 369	. 2 |- (((ph -> -. ps) /\ (-. ps -> ph)) <-> ((ps -> -. ph) /\ (-. ph -> ps)))
4		bi 396	. 2 |- ((ph <-> -. ps) <-> ((ph -> -. ps) /\ (-. ps -> ph)))
5		bi 396	. 2 |- ((ps <-> -. ph) <-> ((ps -> -. ph) /\ (-. ph -> ps)))
6	3, 4, 5	3bitr4 158	1 |- ((ph <-> -. ps) <-> (ps <-> -. ph))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  bicon2d 404

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198

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