Theorem syl6bir 220

Description: A mixed syllogism inference. (Contributed by NM, 18-May-1994.)

Hypotheses

Ref	Expression
syl6bir.1	⊢ (φ → (χ ↔ ψ))
syl6bir.2	⊢ (χ → θ)

Assertion

Ref	Expression
syl6bir	⊢ (φ → (ψ → θ))

Proof of Theorem syl6bir

Step	Hyp	Ref	Expression
1		syl6bir.1	. . 3 ⊢ (φ → (χ ↔ ψ))
2	1	biimprd 214	. 2 ⊢ (φ → (ψ → χ))
3		syl6bir.2	. 2 ⊢ (χ → θ)
4	2, 3	syl6 29	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:  19.21t  1795  exdistrf  1971  ax11  2155  fnun  5190  ovigg  5597  fvmpti  5700  ce0addcnnul  6180  ce0nnulb  6183  ceclb  6184