Theorem axc14 2456

Description: Axiom ax-c14 38490 is redundant if we assume ax-5 1905. 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 2455 and ax-5 1905. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by NM, 29-Jun-1995.) (Proof modification is discouraged.) (New usage 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 2130	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 ¬ ∀𝑧 𝑧 = 𝑦)
2		dveel2 2455	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑤 ∈ 𝑦 → ∀𝑧 𝑤 ∈ 𝑦))
3	1, 2	hbim1 2286	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦))
4		elequ1 2105	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
5	4	imbi2d 339	. . . 4 ⊢ (𝑤 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑤 ∈ 𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦)))
6	3, 5	dvelim 2444	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → ((¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦) → ∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦)))
7		nfa1 2140	. . . . 5 ⊢ Ⅎ𝑧∀𝑧 𝑧 = 𝑦
8	7	nfn 1852	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
9	8	19.21 2195	. . 3 ⊢ (∀𝑧(¬ ∀𝑧 𝑧 = 𝑦 → 𝑥 ∈ 𝑦) ↔ (¬ ∀𝑧 𝑧 = 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))
10	6, 9	imbitrdi 250	. 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 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2365

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778

This theorem is referenced by: (None)