Theorem pm2.61ddan 826

Description: Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.)

Hypotheses

Ref	Expression
pm2.61ddan.1	⊢ ((𝜑 ∧ 𝜓) → 𝜃)
pm2.61ddan.2	⊢ ((𝜑 ∧ 𝜒) → 𝜃)
pm2.61ddan.3	⊢ ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)

Assertion

Ref	Expression
pm2.61ddan	⊢ (𝜑 → 𝜃)

Proof of Theorem pm2.61ddan

Step	Hyp	Ref	Expression
1		pm2.61ddan.1	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
2		pm2.61ddan.2	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)
3	2	adantlr 728	. . 3 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ 𝜒) → 𝜃)
4		pm2.61ddan.3	. . . 4 ⊢ ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃)
5	4	anassrs 473	. . 3 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ ¬ 𝜒) → 𝜃)
6	3, 5	pm2.61dan 825	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃)
7	1, 6	pm2.61dan 825	1 ⊢ (𝜑 → 𝜃)

This proof depends on syntax axioms:  ¬ wn 3   → 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:  ecase3ad  1052  lgsdir2  27545  cdlemg24  41520  initopropd  50078  termopropd  50079  zeroopropd  50080