Theorem pm2.61ddan 767

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 695	. . 3 ⊢ (((φ ∧ ¬ ψ) ∧ χ) → θ)
4		pm2.61ddan.3	. . . 4 ⊢ ((φ ∧ (¬ ψ ∧ ¬ χ)) → θ)
5	4	anassrs 629	. . 3 ⊢ (((φ ∧ ¬ ψ) ∧ ¬ χ) → θ)
6	3, 5	pm2.61dan 766	. 2 ⊢ ((φ ∧ ¬ ψ) → θ)
7	1, 6	pm2.61dan 766	1 ⊢ (φ → θ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358

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 177  df-an 360

This theorem is referenced by: (None)