Theorem bi3 132

Description: Property of the biconditional connective.

Assertion

Ref	Expression
bi3	|- ((ph -> ps) -> ((ps -> ph) -> (ph <-> ps)))

Proof of Theorem bi3

Step	Hyp	Ref	Expression
1		df-bi 128	. . 3 |- -. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
2		pm3.27im 121	. . 3 |- (-. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))) -> (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
3	1, 2	ax-mp 6	. 2 |- (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))
4	3	expi 125	1 |- ((ph -> ps) -> ((ps -> ph) -> (ph <-> ps)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127

This theorem is referenced by:  impbi 139  bii 140

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

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