Theorem pm10.57 37592

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

Assertion

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

Proof of Theorem pm10.57

Step	Hyp	Ref	Expression
1		alnex 1697	. . . 4 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜒) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜒))
2		imnan 437	. . . . . 6 ⊢ ((𝜑 → ¬ 𝜒) ↔ ¬ (𝜑 ∧ 𝜒))
3		pm2.53 387	. . . . . . . 8 ⊢ ((𝜓 ∨ 𝜒) → (¬ 𝜓 → 𝜒))
4	3	con1d 138	. . . . . . 7 ⊢ ((𝜓 ∨ 𝜒) → (¬ 𝜒 → 𝜓))
5	4	imim3i 62	. . . . . 6 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → ((𝜑 → ¬ 𝜒) → (𝜑 → 𝜓)))
6	2, 5	syl5bir 232	. . . . 5 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → (¬ (𝜑 ∧ 𝜒) → (𝜑 → 𝜓)))
7	6	al2imi 1733	. . . 4 ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥 ¬ (𝜑 ∧ 𝜒) → ∀𝑥(𝜑 → 𝜓)))
8	1, 7	syl5bir 232	. . 3 ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (¬ ∃𝑥(𝜑 ∧ 𝜒) → ∀𝑥(𝜑 → 𝜓)))
9	8	con1d 138	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (¬ ∀𝑥(𝜑 → 𝜓) → ∃𝑥(𝜑 ∧ 𝜒)))
10	9	orrd 392	1 ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ∨ ∃𝑥(𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696

This theorem is referenced by: (None)