Theorem syl2anr 464

Description: A double syllogism inference. (Contributed by NM, 17-Sep-2013.)

Hypotheses

Ref	Expression
syl2an.1	⊢ (φ → ψ)
syl2an.2	⊢ (τ → χ)
syl2an.3	⊢ ((ψ ∧ χ) → θ)

Assertion

Ref	Expression
syl2anr	⊢ ((τ ∧ φ) → θ)

Proof of Theorem syl2anr

Step	Hyp	Ref	Expression
1		syl2an.1	. . 3 ⊢ (φ → ψ)
2		syl2an.2	. . 3 ⊢ (τ → χ)
3		syl2an.3	. . 3 ⊢ ((ψ ∧ χ) → θ)
4	1, 2, 3	syl2an 463	. 2 ⊢ ((φ ∧ τ) → θ)
5	4	ancoms 439	1 ⊢ ((τ ∧ φ) → θ)

Syntax hints:   → wi 4   ∧ 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:  funco  5143  resdif  5307  enmap2lem5  6068  sbthlem3  6206