Theorem sylibd 205

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

Hypotheses

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

Assertion

Ref	Expression
sylibd	⊢ (φ → (ψ → θ))

Proof of Theorem sylibd

Step	Hyp	Ref	Expression
1		sylibd.1	. 2 ⊢ (φ → (ψ → χ))
2		sylibd.2	. . 3 ⊢ (φ → (χ ↔ θ))
3	2	biimpd 198	. 2 ⊢ (φ → (χ → θ))
4	1, 3	syld 40	1 ⊢ (φ → (ψ → θ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  3imtr3d  258  ceqsalt  2882  sbceqal  3098  sbcimdv  3108  csbiebt  3173  rspcsbela  3196  ltfintri  4467  copsexg  4608  fce  6189  spacind  6288