Theorem sylibrd 225

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

Hypotheses

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

Assertion

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

Proof of Theorem sylibrd

Step	Hyp	Ref	Expression
1		sylibrd.1	. 2 ⊢ (φ → (ψ → χ))
2		sylibrd.2	. . 3 ⊢ (φ → (θ ↔ χ))
3	2	biimprd 214	. 2 ⊢ (φ → (χ → θ))
4	1, 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:  3imtr4d  259  sbciegft  3076  eqsbc3r  3103  fconstfv  5456  nmembers1  6271  nchoicelem12  6300