Theorem a1bi 355

Description: Inference 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 353	. 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (𝜓 ↔ (𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 198

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 199

This theorem is referenced by:  mt2bi  356  pm4.83  1008  trut  1514  equsv  1961  equsalv  2197  equsal  2353  2sb6rf  2421  sb4b  2425  sbequ8  2469  ralv  3442  relop  5575  acsfn0  16801  cmpsub  21727  ballotlemodife  31433  bj-sbievw1  33700  bj-sbievw  33702  bj-ralvw  33728  wl-equsalvw  34260  wl-equsald  34261  lub0N  35810  glb0N  35814