Theorem ccase 912

Description: Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)

Hypotheses

Ref	Expression
ccase.1	⊢ ((φ ∧ ψ) → τ)
ccase.2	⊢ ((χ ∧ ψ) → τ)
ccase.3	⊢ ((φ ∧ θ) → τ)
ccase.4	⊢ ((χ ∧ θ) → τ)

Assertion

Ref	Expression
ccase	⊢ (((φ ∨ χ) ∧ (ψ ∨ θ)) → τ)

Proof of Theorem ccase

Step	Hyp	Ref	Expression
1		ccase.1	. . 3 ⊢ ((φ ∧ ψ) → τ)
2		ccase.2	. . 3 ⊢ ((χ ∧ ψ) → τ)
3	1, 2	jaoian 759	. 2 ⊢ (((φ ∨ χ) ∧ ψ) → τ)
4		ccase.3	. . 3 ⊢ ((φ ∧ θ) → τ)
5		ccase.4	. . 3 ⊢ ((χ ∧ θ) → τ)
6	4, 5	jaoian 759	. 2 ⊢ (((φ ∨ χ) ∧ θ) → τ)
7	3, 6	jaodan 760	1 ⊢ (((φ ∨ χ) ∧ (ψ ∨ θ)) → τ)

Syntax hints:   → wi 4   ∨ wo 357   ∧ 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-or 359  df-an 360

This theorem is referenced by:  ccased  913  ccase2  914  undif3  3516