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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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 176

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