Theorem a1bi 327

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 325	. 2 ⊢ (φ → (ψ ↔ (φ → ψ)))
3	1, 2	ax-mp 5	1 ⊢ (ψ ↔ (φ → ψ))

Syntax hints:   → wi 4   ↔ wb 176

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 177

This theorem is referenced by:  mt2bi  328  pm4.83  895  truimfal  1345  equsalhw  1838  equveli  1988  sbequ8  2079  ralv  2872  ssopr  4846