Theorem pm2.61dda 813

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

Hypotheses

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

Assertion

Ref	Expression
pm2.61dda	⊢ (𝜑 → 𝜃)

Proof of Theorem pm2.61dda

Step	Hyp	Ref	Expression
1		pm2.61dda.3	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
2	1	anassrs 470	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
3		pm2.61dda.2	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜒) → 𝜃)
4	3	adantlr 713	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ ¬ 𝜒) → 𝜃)
5	2, 4	pm2.61dan 811	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
6		pm2.61dda.1	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃)
7	5, 6	pm2.61dan 811	1 ⊢ (𝜑 → 𝜃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  lhpexle1lem  37135  lclkrlem2x  38658