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  3730  eusv2i  4599  ffdm  5556  ov  6201  ovg  6221  nnacl  6746  elpm2r  6933  ltnqpri  7954  ltxrlt  8384  uzaddcl  9968  fzspl  10457  expcllem  10968  qexpclz  10978  1exp  10986  facnn  11146  fac0  11147  fac1  11148  bcn2  11183  en1hash  11220  hash2en  11276  znnen  13270  zrhval  14927