Theorem axextdist 36654

Description: ax-ext 2732 with distinctors instead of distinct variable conditions. (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 2463	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2		nfnae 2463	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
3	1, 2	nfan 1931	. . 3 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4		nfcvf 2948	. . . . . 6 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧𝑥)
5	4	adantr 486	. . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧𝑥)
6	5	nfcrd 2916	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤 ∈ 𝑥)
7		nfcvf 2948	. . . . . 6 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝑦)
8	7	adantl 487	. . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧𝑦)
9	8	nfcrd 2916	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤 ∈ 𝑦)
10	6, 9	nfbid 1934	. . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦))
11		elequ1 2151	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
12		elequ1 2151	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
13	11, 12	bibi12d 348	. . . 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 2436	. 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
16		axextg 2734	. 2 ⊢ (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)
17	15, 16	biimtrrdi 257	1 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  Ⅎwnfc 2907

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-clel 2835  df-nfc 2909

This theorem is used by:  axextbdist  36655