Theorem ccase2 564

Description: Inference for combining cases.

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 306	. 2 ⊢ ((χ ∧ ψ) → τ)
4		ccase2.3	. . 3 ⊢ (θ → τ)
5	4	adantl 305	. 2 ⊢ ((φ ∧ θ) → τ)
6	4	adantl 305	. 2 ⊢ ((χ ∧ θ) → τ)
7	1, 3, 5, 6	ccase 562	1 ⊢ (((φ ∨ χ) ∧ (ψ ∨ θ)) → τ)

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  add20 4329  mulge0 4335  nn0mulcl 4553

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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