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  3695  eusv2i  4558  ffdm  5513  ov  6151  ovg  6171  nnacl  6691  elpm2r  6878  ltnqpri  7857  ltxrlt  8287  uzaddcl  9864  expcllem  10858  qexpclz  10868  1exp  10876  facnn  11035  fac0  11036  fac1  11037  bcn2  11072  en1hash  11108  hash2en  11153  znnen  13082  zrhval  14696