Theorem ibir 176

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 140	. 2 ⊢ (𝜑 → (𝜑 ↔ 𝜓))
3	2	ibi 175	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.21nii  694  elpr2  3582  eusv2i  4413  ffdm  5337  ov  5934  ovg  5953  nnacl  6420  elpm2r  6604  ltnqpri  7497  ltxrlt  7926  uzaddcl  9480  expcllem  10412  qexpclz  10422  1exp  10430  facnn  10583  fac0  10584  fac1  10585  bcn2  10620  znnen  12099