Theorem mpanlr1 719

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 533	. 2 ⊢ (𝜒 → (𝜓 ∧ 𝜒))
3		mpanlr1.2	. 2 ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)
4	2, 3	sylanl2 694	1 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  oecl  8528  omass  8571  oen0  8578  oeordi  8579  oewordri  8584  oeworde  8585