Theorem syl2anbr 466

Description: A double syllogism inference. (Contributed by NM, 29-Jul-1999.)

Hypotheses

Ref	Expression
syl2anbr.1	⊢ (ψ ↔ φ)
syl2anbr.2	⊢ (χ ↔ τ)
syl2anbr.3	⊢ ((ψ ∧ χ) → θ)

Assertion

Ref	Expression
syl2anbr	⊢ ((φ ∧ τ) → θ)

Proof of Theorem syl2anbr

Step	Hyp	Ref	Expression
1		syl2anbr.2	. 2 ⊢ (χ ↔ τ)
2		syl2anbr.1	. . 3 ⊢ (ψ ↔ φ)
3		syl2anbr.3	. . 3 ⊢ ((ψ ∧ χ) → θ)
4	2, 3	sylanbr 459	. 2 ⊢ ((φ ∧ χ) → θ)
5	1, 4	sylan2br 462	1 ⊢ ((φ ∧ τ) → 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:  sylancbr  647  ov3  5600