Theorem 4cases 1056

Description: Inference eliminating two antecedents from the four possible cases that result from their true/false combinations. (Contributed by NM, 25-Oct-2003.)

Hypotheses

Ref	Expression
4cases.1	⊢ ((𝜑 ∧ 𝜓) → 𝜒)
4cases.2	⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)
4cases.3	⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒)
4cases.4	⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)

Assertion

Ref	Expression
4cases	⊢ 𝜒

Proof of Theorem 4cases

Step	Hyp	Ref	Expression
1		4cases.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
2		4cases.3	. . 3 ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒)
3	1, 2	pm2.61ian 824	. 2 ⊢ (𝜓 → 𝜒)
4		4cases.2	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒)
5		4cases.4	. . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)
6	4, 5	pm2.61ian 824	. 2 ⊢ (¬ 𝜓 → 𝜒)
7	3, 6	pm2.61i 184	1 ⊢ 𝜒

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  4casesdan  1057  suc11reg  9595  hasheqf1oi  14405  fvprmselgcd1  17127  axlowdimlem15  29361  ax12eq  39773  ax12el  39774  cdleme27a  41199