Theorem a1bi 242

Description: Inference introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)

Hypothesis

Ref	Expression
a1bi.1	|- ph

Assertion

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

Proof of Theorem a1bi

Step	Hyp	Ref	Expression
1		a1bi.1	. 2 |- ph
2		biimt 240	. 2 |- ( ph -> ( ps <-> ( ph -> ps ) ) )
3	1, 2	ax-mp 5	1 |- ( ps <-> ( ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  mt2bi  674  truimfal  1400  equsal  1715  equveli  1747  equsalv  1781  ralv  2743  relop  4754