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  6559  distrlem4pru  6560  recexprlemm  6595  aptisr  6685  renegcl  7048  remulext2  7364  mulext1  7376  apmul1  7526  nnge1  7698  0mnnnnn0  7970  nn0lt2  8078  zneo  8095  uzind2  8106  fzind  8109  nn0ind-raph  8111  elfzmlbp  8740  difelfznle  8743  elfzodifsumelfzo  8807  ssfzo12  8830