Theorem a1bi 351

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	⊢ 𝜑

Assertion

Ref	Expression
a1bi	⊢ (𝜓 ↔ (𝜑 → 𝜓))

Proof of Theorem a1bi

Step	Hyp	Ref	Expression
1		a1bi.1	. 2 ⊢ 𝜑
2		biimt 349	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 195

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 196

This theorem is referenced by:  mt2bi  352  pm4.83  966  truimfal  1506  equsalvw  1918  equsalhw  2109  equsal  2279  sbequ8ALT  2395  ralv  3192  relop  5194  acsfn0  16144  cmpsub  21013  ballotlemodife  29886  bj-trut  31740  bj-ssb1  31822  bj-equsalv  31931  bj-ralvw  32059  wl-equsald  32504  lub0N  33494  glb0N  33498