Theorem ccase2 1040

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 480	. 2 ⊢ ((𝜒 ∧ 𝜓) → 𝜏)
4		ccase2.3	. . 3 ⊢ (𝜃 → 𝜏)
5	4	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝜃) → 𝜏)
6	4	adantl 481	. 2 ⊢ ((𝜒 ∧ 𝜃) → 𝜏)
7	1, 3, 5, 6	ccase 1038	1 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜃)) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848

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 207  df-an 396  df-or 849

This theorem is referenced by:  opthhausdorff  5522  fctop  23011  cctop  23013