Theorem sylibd 138

Description: A syllogism deduction. (Contributed by NM, 3-Aug-1994.)

Hypotheses

Ref	Expression
sylibd.1	⊢ (φ → (ψ → χ))
sylibd.2	⊢ (φ → (χ ↔ θ))

Assertion

Ref	Expression
sylibd	⊢ (φ → (ψ → θ))

Proof of Theorem sylibd

Step	Hyp	Ref	Expression
1		sylibd.1	. 2 ⊢ (φ → (ψ → χ))
2		sylibd.2	. . 3 ⊢ (φ → (χ ↔ θ))
3	2	biimpd 132	. 2 ⊢ (φ → (χ → θ))
4	1, 3	syld 40	1 ⊢ (φ → (ψ → θ))

Syntax hints:   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  3imtr3d  191  dvelimdf  1889  ceqsalt  2574  sbceqal  2808  sbcimdv  2817  csbiebt  2880  rspcsbela  2899  preqr1g  3528  repizf2  3906  copsexg  3972  onun2  4182  suc11g  4235  elrnrexdm  5249  isoselem  5402  riotass2  5437  nnm00  6038  ecopovtrn  6139  ecopovtrng  6142  enq0tr  6416  addnqprl  6512  addnqpru  6513  mulnqprl  6547  mulnqpru  6548  recexprlemss1l  6605  recexprlemss1u  6606  mulextsr1lem  6666  pitonn  6704  cnegexlem1  6943  ltadd2  7172  mulext  7358  mulgt1  7570  lt2halves  7897  addltmul  7898  zextlt  8068  recnz  8069  zeo  8079  peano5uzti  8082  irradd  8315  irrmul  8316  xltneg  8479  icc0r  8525  fznuz  8694  uznfz  8695  bj-peano4  9343