Theorem a1bi 197

Description: Inference rule introducing a theorem as an antecedent.

Hypothesis

Ref	Expression
a1bi.1	|- ph

Assertion

Ref	Expression
a1bi	|- (ps <-> (ph -> ps))

Proof of Theorem a1bi

Step	Hyp	Ref	Expression
1		ax-1 4	. 2 |- (ps -> (ph -> ps))
2		a1bi.1	. . 3 |- ph
3		pm2.27 62	. . 3 |- (ph -> ((ph -> ps) -> ps))
4	2, 3	ax-mp 7	. 2 |- ((ph -> ps) -> ps)
5	1, 4	impbi 157	1 |- (ps <-> (ph -> ps))

Syntax hints:   -> wi 3   <-> wb 146

This theorem is referenced by:  pm4.83 740  sbequ8 1247  a12lem1 1376  ralv 1820  hbsbc1v 1950  relop 3275  pw2en 4446  caun0 7945

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

Copyright terms: Public domain