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  709  elpr2  3689  eusv2i  4550  ffdm  5502  ov  6136  ovg  6156  nnacl  6643  elpm2r  6830  ltnqpri  7804  ltxrlt  8235  uzaddcl  9810  expcllem  10802  qexpclz  10812  1exp  10820  facnn  10979  fac0  10980  fac1  10981  bcn2  11016  en1hash  11052  hash2en  11097  znnen  13009  zrhval  14621