Theorem axc14 2400

Description: Axiom ax-c14 35478 is redundant if we assume ax-5 1869. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'.

Note that 𝑤 is a dummy variable introduced in the proof. Its purpose is to satisfy the distinct variable requirements of dveel2 2399 and ax-5 1869. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc14	⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))

Proof of Theorem axc14

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbn1 2080	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 ¬ ∀𝑧 𝑧 = 𝑦)
2		dveel2 2399	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑤 ∈ 𝑦 → ∀𝑧 𝑤 ∈ 𝑦))
3	1, 2	hbim1 2231	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦))
4		elequ1 2057	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
5	4	imbi2d 333	. . . 4 ⊢ (𝑤 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦)))
6	3, 5	dvelim 2387	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦)))
7		nfa1 2088	. . . . 5 ⊢ Ⅎ𝑧∀𝑧 𝑧 = 𝑦
8	7	nfn 1819	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
9	8	19.21 2136	. . 3 ⊢ (∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))
10	6, 9	syl6ib 243	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
11	10	pm2.86d 108	1 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747

This theorem is referenced by: (None)