Theorem axextdist 36032

Description: ax-ext 2712 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 2442	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑥
2		nfnae 2442	. . . 4 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
3	1, 2	nfan 1906	. . 3 ⊢ Ⅎ𝑧(¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦)
4		nfcvf 2928	. . . . . 6 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧𝑥)
5	4	adantr 481	. . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧𝑥)
6	5	nfcrd 2896	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤 ∈ 𝑥)
7		nfcvf 2928	. . . . . 6 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝑦)
8	7	adantl 482	. . . . 5 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧𝑦)
9	8	nfcrd 2896	. . . 4 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑤 ∈ 𝑦)
10	6, 9	nfbid 1909	. . 3 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦))
11		elequ1 2126	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
12		elequ1 2126	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦))
13	11, 12	bibi12d 346	. . . 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 2415	. 2 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
16		axextg 2714	. 2 ⊢ (∀𝑤(𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦) → 𝑥 = 𝑦)
17	15, 16	biimtrrdi 255	1 ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  Ⅎwnfc 2887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-clel 2815  df-nfc 2889

This theorem is referenced by:  axextbdist  36033