Theorem bi3 166

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 163	. . 3 |- -. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
2		pm3.27im 154	. . 3 |- (-. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))) -> (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
3	1, 2	ax-mp 7	. 2 |- (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))
4	3	expi 160	1 |- ((ph -> ps) -> ((ps -> ph) -> (ph <-> ps)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 162

This theorem is referenced by:  impbii 173  dfbi1 174  asymref2OLD 4122  eqsbc3rVD 16323  orbi1rVD 16331  3impexpVD 16339  3impexpbicomVD 16340

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 163

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