Theorem ccase 562

Description: Inference for combining cases.

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		caselem 561	. 2 ⊢ (((φ ∨ χ) ∧ (ψ ∨ θ)) ↔ (((φ ∧ ψ) ∨ (χ ∧ ψ)) ∨ ((φ ∧ θ) ∨ (χ ∧ θ))))
2		ccase.1	. . . 4 ⊢ ((φ ∧ ψ) → τ)
3		ccase.2	. . . 4 ⊢ ((χ ∧ ψ) → τ)
4	2, 3	jaoi 275	. . 3 ⊢ (((φ ∧ ψ) ∨ (χ ∧ ψ)) → τ)
5		ccase.3	. . . 4 ⊢ ((φ ∧ θ) → τ)
6		ccase.4	. . . 4 ⊢ ((χ ∧ θ) → τ)
7	5, 6	jaoi 275	. . 3 ⊢ (((φ ∧ θ) ∨ (χ ∧ θ)) → τ)
8	4, 7	jaoi 275	. 2 ⊢ ((((φ ∧ ψ) ∨ (χ ∧ ψ)) ∨ ((φ ∧ θ) ∨ (χ ∧ θ))) → τ)
9	1, 8	sylbi 174	1 ⊢ (((φ ∨ χ) ∧ (ψ ∨ θ)) → τ)

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

This theorem is referenced by:  ccase2 564  addge0 4324  lt2sq 4414  nn0addclt 4551  nn0ltp1let 4556

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

metamath.org