Theorem sylbird 226

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 214	. 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 176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  3imtr3d  258  eqreu  3028  sfinltfin  4535  ov3  5599  erref  5959  enmap2lem3  6065  nenpw1pwlem2  6085  tlecg  6230  ce2le  6233  addcdi  6250  nchoicelem9  6297