Theorem sylbird 159

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

Hypotheses

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

Assertion

Ref	Expression
sylbird	⊢ (φ → (ψ → θ))

Proof of Theorem sylbird

Step	Hyp	Ref	Expression
1		sylbird.1	. . 3 ⊢ (φ → (χ ↔ ψ))
2	1	biimprd 147	. 2 ⊢ (φ → (ψ → χ))
3		sylbird.2	. 2 ⊢ (φ → (χ → θ))
4	2, 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  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  3imtr3d  191  drex1  1676  eqreu  2727  onsucsssucr  4200  ordsucunielexmid  4216  ovi3  5579  tfrlem9  5876  rdgon  5913  dom2lem  6188  distrlem4prl  6558  distrlem4pru  6559  recexprlemm  6594  aptisr  6665  renegcl  7028  remulext2  7344  mulext1  7356  apmul1  7506  nnge1  7678  0mnnnnn0  7950  nn0lt2  8058  zneo  8075  uzind2  8086  fzind  8089  nn0ind-raph  8091  elfzmlbp  8720  difelfznle  8723  elfzodifsumelfzo  8787  ssfzo12  8810