Theorem pm2.61dda 768

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