Theorem a1bi 350

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 348	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 194

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 195

This theorem is referenced by:  mt2bi  351  pm4.83  965  truimfal  1505  equsalvw  1917  equsalhw  2108  equsal  2278  sbequ8ALT  2394  ralv  3191  relop  5182  acsfn0  16090  cmpsub  20955  ballotlemodife  29692  bj-trut  31546  bj-ssb1  31628  bj-equsalv  31737  bj-ralvw  31855  wl-equsald  32300  lub0N  33290  glb0N  33294