Theorem bii 158

Description: Relate the biconditional connective to primitive connectives. See biigb 159 for an unusual version proved directly from axioms.

Assertion

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

Proof of Theorem bii

Step	Hyp	Ref	Expression
1		bi1 148	. . 3 |- ((ph <-> ps) -> (ph -> ps))
2		bi2 149	. . 3 |- ((ph <-> ps) -> (ps -> ph))
3	1, 2	jc 138	. 2 |- ((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph)))
4		bi3 150	. . 3 |- ((ph -> ps) -> ((ps -> ph) -> (ph <-> ps)))
5	4	impi 143	. 2 |- (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))
6	3, 5	impbi 157	1 |- ((ph <-> ps) <-> -. ((ph -> ps) -> -. (ps -> ph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146

This theorem is referenced by:  bi 513

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

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