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 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  352  pm4.83  990  trut  1532  equsalvw  1977  equsalv  2146  equsalhw  2161  equsal  2327  sbequ8ALT  2435  ralv  3250  relop  5305  acsfn0  16368  cmpsub  21251  ballotlemodife  30687  bj-ssb1  32758  bj-ralvw  32990  wl-equsald  33455  lub0N  34794  glb0N  34798