Theorem bi2 131

Description: Property of the biconditional connective.

Assertion

Ref	Expression
bi2	|- ((ph <-> ps) -> (ps -> ph))

Proof of Theorem bi2

Step	Hyp	Ref	Expression
1		df-bi 128	. . 3 |- -. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps)))
2		pm3.26im 120	. . 3 |- (-. (((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))) -> -. (-. ((ph -> ps) -> -. (ps -> ph)) -> (ph <-> ps))) -> ((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph))))
3	1, 2	ax-mp 6	. 2 |- ((ph <-> ps) -> -. ((ph -> ps) -> -. (ps -> ph)))
4		pm3.27im 121	. 2 |- (-. ((ph -> ps) -> -. (ps -> ph)) -> (ps -> ph))
5	3, 4	syl 12	1 |- ((ph <-> ps) -> (ps -> ph))

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

This theorem is referenced by:  biimpr 134  biimprd 136  bii 140  pm5.74 442  pm4.72 485  tbt 541  pclem6 555  19.15 694  19.18 732  cbv2 846  sbied 903  fv3 2839

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

metamath.org