Theorem sylanbr 459

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

Hypotheses

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

Assertion

Ref	Expression
sylanbr	⊢ ((φ ∧ χ) → θ)

Proof of Theorem sylanbr

Step	Hyp	Ref	Expression
1		sylanbr.1	. . 3 ⊢ (ψ ↔ φ)
2	1	biimpri 197	. 2 ⊢ (φ → ψ)
3		sylanbr.2	. 2 ⊢ ((ψ ∧ χ) → θ)
4	2, 3	sylan 457	1 ⊢ ((φ ∧ χ) → θ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

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  df-an 360

This theorem is referenced by:  syl2anbr  466  funfv  5376