Theorem mpanlr1 438

Description: An inference based on modus ponens. (Contributed by NM, 30-Dec-2004.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

Ref	Expression
mpanlr1.1	⊢ 𝜓
mpanlr1.2	⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
mpanlr1	⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Proof of Theorem mpanlr1

Step	Hyp	Ref	Expression
1		mpanlr1.1	. . 3 ⊢ 𝜓
2	1	jctl 312	. 2 ⊢ (𝜒 → (𝜓 ∧ 𝜒))
3		mpanlr1.2	. 2 ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)
4	2, 3	sylanl2 401	1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem is referenced by: (None)