Theorem pm10.57

Description: Theorem *10.57 in WhiteheadRussell p. 156. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.57	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∨ ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		alnex	⊢ ( ∀ 𝑥 ¬ ( 𝜑 ∧ 𝜒 ) ↔ ¬ ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) )
2		imnan	⊢ ( ( 𝜑 → ¬ 𝜒 ) ↔ ¬ ( 𝜑 ∧ 𝜒 ) )
3		pm2.53	⊢ ( ( 𝜓 ∨ 𝜒 ) → ( ¬ 𝜓 → 𝜒 ) )
4	3	con1d	⊢ ( ( 𝜓 ∨ 𝜒 ) → ( ¬ 𝜒 → 𝜓 ) )
5	4	imim3i	⊢ ( ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) → ( ( 𝜑 → ¬ 𝜒 ) → ( 𝜑 → 𝜓 ) ) )
6	2 5	syl5bir	⊢ ( ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) → ( ¬ ( 𝜑 ∧ 𝜒 ) → ( 𝜑 → 𝜓 ) ) )
7	6	al2imi	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) → ( ∀ 𝑥 ¬ ( 𝜑 ∧ 𝜒 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
8	1 7	syl5bir	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) → ( ¬ ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) → ∀ 𝑥 ( 𝜑 → 𝜓 ) ) )
9	8	con1d	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) → ( ¬ ∀ 𝑥 ( 𝜑 → 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) ) )
10	9	orrd	⊢ ( ∀ 𝑥 ( 𝜑 → ( 𝜓 ∨ 𝜒 ) ) → ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∨ ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) ) )