Theorem ax6vsep 5021

Description: Derive ax6v 2022 (a weakened version of ax-6 2021 where 𝑥 and 𝑦 must be distinct), from Separation ax-sep 5017 and Extensionality ax-ext 2754. See ax6 2348 for the derivation of ax-6 2021 from ax6v 2022. (Contributed by NM, 12-Nov-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6vsep	⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem ax6vsep

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 5017	. . 3 ⊢ ∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧)))
2		id 22	. . . . . . . . 9 ⊢ (𝑧 = 𝑧 → 𝑧 = 𝑧)
3	2	biantru 525	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧)))
4	3	bibi2i 329	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))))
5	4	biimpri 220	. . . . . 6 ⊢ ((𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))) → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
6	5	alimi 1855	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))) → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
7		ax-ext 2754	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
8	6, 7	syl 17	. . . 4 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))) → 𝑥 = 𝑦)
9	8	eximi 1878	. . 3 ⊢ (∃𝑥∀𝑧(𝑧 ∈ 𝑥 ↔ (𝑧 ∈ 𝑦 ∧ (𝑧 = 𝑧 → 𝑧 = 𝑧))) → ∃𝑥 𝑥 = 𝑦)
10	1, 9	ax-mp 5	. 2 ⊢ ∃𝑥 𝑥 = 𝑦
11		df-ex 1824	. 2 ⊢ (∃𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 ¬ 𝑥 = 𝑦)
12	10, 11	mpbi 222	1 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599   = wceq 1601  ∃wex 1823   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-ext 2754  ax-sep 5017

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824

This theorem is referenced by: (None)