Theorem falseral0OLD 4466

Description: Obsolete version of falseral0 4465 as of 16-Feb-2026. (Contributed by AV, 30-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
falseral0OLD	⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem falseral0OLD

Step	Hyp	Ref	Expression
1		df-ral 3076	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		19.26 1889	. . 3 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) ↔ (∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)))
3		con3 153	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (¬ 𝜑 → ¬ 𝑥 ∈ 𝐴))
4	3	impcom 411	. . . . . 6 ⊢ ((¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ¬ 𝑥 ∈ 𝐴)
5	4	alimi 1830	. . . . 5 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ∀𝑥 ¬ 𝑥 ∈ 𝐴)
6		alnex 1800	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
7	5, 6	sylib 220	. . . 4 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → ¬ ∃𝑥 𝑥 ∈ 𝐴)
8		notnotb 317	. . . . 5 ⊢ (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
9		neq0 4302	. . . . 5 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
10	8, 9	xchbinx 336	. . . 4 ⊢ (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
11	7, 10	sylibr 236	. . 3 ⊢ (∀𝑥(¬ 𝜑 ∧ (𝑥 ∈ 𝐴 → 𝜑)) → 𝐴 = ∅)
12	2, 11	sylbir 237	. 2 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) → 𝐴 = ∅)
13	1, 12	sylan2b 603	1 ⊢ ((∀𝑥 ¬ 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜑) → 𝐴 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  ∅c0 4283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-ral 3076  df-dif 3905  df-nul 4284

This theorem is referenced by: (None)