Theorem axextdist 31705

Description: ax-ext 2602 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)

Assertion

Ref	Expression
axextdist	⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))

Proof of Theorem axextdist

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnae 2318	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2		nfnae 2318	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
3	1, 2	nfan 1828	. . 3 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4		nfcvf 2788	. . . . . 6 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧𝑥)
5	4	adantr 481	. . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧𝑥)
6	5	nfcrd 2771	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤 ∈ 𝑥)
7		nfcvf 2788	. . . . . 6 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝑦)
8	7	adantl 482	. . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧𝑦)
9	8	nfcrd 2771	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤 ∈ 𝑦)
10	6, 9	nfbid 1832	. . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦))
11		elequ1 1997	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
12		elequ1 1997	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
13	11, 12	bibi12d 335	. . . 4 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
14	13	a1i 11	. . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑤 = 𝑧 → ((𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))))
15	3, 10, 14	cbvald 2277	. 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
16		axext3 2604	. 2 ⊢ (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)
17	15, 16	syl6bir 244	1 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  axext4dist  31706