Theorem a1bi 352

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 350	. 2 |- ( ph -> ( ps <-> ( ph -> ps ) ) )
3	1, 2	ax-mp 5	1 |- ( ps <-> ( ph -> ps ) )

Syntax hints:   -> wi 4   <-> wb 196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  mt2bi  353  pm4.83  970  trut  1492  equsalvw  1931  equsalv  2108  equsalhw  2123  equsal  2291  sbequ8ALT  2407  ralv  3219  relop  5272  acsfn0  16321  cmpsub  21203  ballotlemodife  30559  bj-ssb1  32633  bj-ralvw  32865  wl-equsald  33325  lub0N  34476  glb0N  34480