Theorem 4cases 915

Description: Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.)

Hypotheses

Ref	Expression
4cases.1	⊢ ((φ ∧ ψ) → χ)
4cases.2	⊢ ((φ ∧ ¬ ψ) → χ)
4cases.3	⊢ ((¬ φ ∧ ψ) → χ)
4cases.4	⊢ ((¬ φ ∧ ¬ ψ) → χ)

Assertion

Ref	Expression
4cases	⊢ χ

Proof of Theorem 4cases

Step	Hyp	Ref	Expression
1		4cases.1	. . 3 ⊢ ((φ ∧ ψ) → χ)
2		4cases.3	. . 3 ⊢ ((¬ φ ∧ ψ) → χ)
3	1, 2	pm2.61ian 765	. 2 ⊢ (ψ → χ)
4		4cases.2	. . 3 ⊢ ((φ ∧ ¬ ψ) → χ)
5		4cases.4	. . 3 ⊢ ((¬ φ ∧ ¬ ψ) → χ)
6	4, 5	pm2.61ian 765	. 2 ⊢ (¬ ψ → χ)
7	3, 6	pm2.61i 156	1 ⊢ χ

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:  4casesdan  916  ax11eq  2193  ax11el  2194