Theorem ccase2 914

Description: Inference for combining cases. (Contributed by NM, 29-Jul-1999.)

Hypotheses

Ref	Expression
ccase2.1	⊢ ((φ ∧ ψ) → τ)
ccase2.2	⊢ (χ → τ)
ccase2.3	⊢ (θ → τ)

Assertion

Ref	Expression
ccase2	⊢ (((φ ∨ χ) ∧ (ψ ∨ θ)) → τ)

Proof of Theorem ccase2

Step	Hyp	Ref	Expression
1		ccase2.1	. 2 ⊢ ((φ ∧ ψ) → τ)
2		ccase2.2	. . 3 ⊢ (χ → τ)
3	2	adantr 451	. 2 ⊢ ((χ ∧ ψ) → τ)
4		ccase2.3	. . 3 ⊢ (θ → τ)
5	4	adantl 452	. 2 ⊢ ((φ ∧ θ) → τ)
6	4	adantl 452	. 2 ⊢ ((χ ∧ θ) → τ)
7	1, 3, 5, 6	ccase 912	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: (None)