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  712  elpr2  3716  eusv2i  4581  ffdm  5538  ov  6181  ovg  6201  nnacl  6726  elpm2r  6913  ltnqpri  7925  ltxrlt  8355  uzaddcl  9936  fzspl  10425  expcllem  10936  qexpclz  10946  1exp  10954  facnn  11114  fac0  11115  fac1  11116  bcn2  11151  en1hash  11188  hash2en  11240  znnen  13233  zrhval  14891