Theorem a1bi 232

Description: Inference rule 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 230	. 2 |- ( ph -> ( ps <-> ( ph -> ps ) ) )
3	1, 2	ax-mp 7	1 |- ( ps <-> ( ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  mt2bi  609  truimfal  1301  equsal  1615  equveli  1642  ralv  2571  relop  4486