Theorem pm2.61dda 827

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 473	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
3		pm2.61dda.2	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜒) → 𝜃)
4	3	adantlr 728	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ ¬ 𝜒) → 𝜃)
5	2, 4	pm2.61dan 825	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
6		pm2.61dda.1	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃)
7	5, 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:  tpssad  32958  lhpexle1lem  40841  lclkrlem2x  42364  dmrnxp  49674  initopropd  50080  termopropd  50081  zeroopropd  50082