Theorem pm4.11 522

Description: Contraposition. Theorem *4.11 of [WhiteheadRussell] p. 117.

Assertion

Ref	Expression
pm4.11	|- ((ph <-> ps) <-> (-. ph <-> -. ps))

Proof of Theorem pm4.11

Step	Hyp	Ref	Expression
1		pm4.1 164	. . . 4 |- ((ph -> ps) <-> (-. ps -> -. ph))
2		pm4.1 164	. . . 4 |- ((ps -> ph) <-> (-. ph -> -. ps))
3	1, 2	anbi12i 482	. . 3 |- (((ph -> ps) /\ (ps -> ph)) <-> ((-. ps -> -. ph) /\ (-. ph -> -. ps)))
4		dfbi2 514	. . 3 |- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
5		dfbi2 514	. . 3 |- ((-. ps <-> -. ph) <-> ((-. ps -> -. ph) /\ (-. ph -> -. ps)))
6	3, 4, 5	3bitr4 183	. 2 |- ((ph <-> ps) <-> (-. ps <-> -. ph))
7		bicom 520	. 2 |- ((-. ps <-> -. ph) <-> (-. ph <-> -. ps))
8	6, 7	bitr 173	1 |- ((ph <-> ps) <-> (-. ph <-> -. ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223

This theorem is referenced by:  con4bii 523  con4bid 524  negbid 611  pm5.32 644  nbn2 721  cbvexd 1321

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

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