Theorem ibir 177

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 141	. 2 ⊢ (𝜑 → (𝜑 ↔ 𝜓))
3	2	ibi 176	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm5.21nii  705  elpr2  3645  eusv2i  4491  ffdm  5431  ov  6046  ovg  6066  nnacl  6547  elpm2r  6734  ltnqpri  7680  ltxrlt  8111  uzaddcl  9679  expcllem  10661  qexpclz  10671  1exp  10679  facnn  10838  fac0  10839  fac1  10840  bcn2  10875  znnen  12642  zrhval  14251