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	|- ( ph -> ( ps <-> ph ) )

Assertion

Ref	Expression
ibir	|- ( ph -> ps )

Proof of Theorem ibir

Step	Hyp	Ref	Expression
1		ibir.1	. . 3 |- ( ph -> ( ps <-> ph ) )
2	1	bicomd 141	. 2 |- ( ph -> ( ph <-> ps ) )
3	2	ibi 176	1 |- ( ph -> ps )

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  716  elpr2  3727  eusv2i  4596  ffdm  5553  ov  6198  ovg  6218  nnacl  6743  elpm2r  6930  ltnqpri  7951  ltxrlt  8381  uzaddcl  9965  fzspl  10454  expcllem  10965  qexpclz  10975  1exp  10983  facnn  11143  fac0  11144  fac1  11145  bcn2  11180  en1hash  11217  hash2en  11273  znnen  13267  zrhval  14924