Theorem ibir 233

Description: Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)

Hypothesis

Ref	Expression
ibir.1	⊢ (φ → (ψ ↔ φ))

Assertion

Ref	Expression
ibir	⊢ (φ → ψ)

Proof of Theorem ibir

Step	Hyp	Ref	Expression
1		ibir.1	. . 3 ⊢ (φ → (ψ ↔ φ))
2	1	bicomd 192	. 2 ⊢ (φ → (φ ↔ ψ))
3	2	ibi 232	1 ⊢ (φ → ψ)

Syntax hints:   → wi 4   ↔ wb 176

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  elpr2  3752  opkelimagekg  4271  ffdm  5234  ov  5595  enprmaplem6  6081