Theorem sylan2d 294

Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)

Hypotheses

Ref	Expression
sylan2d.1	⊢ (𝜑 → (𝜓 → 𝜒))
sylan2d.2	⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜏))

Assertion

Ref	Expression
sylan2d	⊢ (𝜑 → ((𝜃 ∧ 𝜓) → 𝜏))

Proof of Theorem sylan2d

Step	Hyp	Ref	Expression
1		sylan2d.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2		sylan2d.2	. . . 4 ⊢ (𝜑 → ((𝜃 ∧ 𝜒) → 𝜏))
3	2	ancomsd 269	. . 3 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝜏))
4	1, 3	syland 293	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) → 𝜏))
5	4	ancomsd 269	1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) → 𝜏))

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  syl2and  295  sylan2i  407  swopo  4342  prarloclemlo  7578  prodgt02  8897  prodge02  8899  infpnlem1  12553