Theorem 2ax6elem 2470

Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. This theorem merges two ax6e 2383 instances ∃𝑧𝑧 = 𝑥 and ∃𝑤𝑤 = 𝑦 into a common expression. Alan Sare contributed a variant of this theorem with distinct variable conditions before, see ax6e2nd 42178. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Wolf Lammen, 27-Sep-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
2ax6elem	⊢ (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))

Proof of Theorem 2ax6elem

Step	Hyp	Ref	Expression
1		ax6e 2383	. . . 4 ⊢ ∃𝑧 𝑧 = 𝑥
2		nfnae 2434	. . . . . 6 ⊢ Ⅎ𝑧 ¬ ∀𝑤 𝑤 = 𝑧
3		nfnae 2434	. . . . . 6 ⊢ Ⅎ𝑧 ¬ ∀𝑤 𝑤 = 𝑥
4	2, 3	nfan 1902	. . . . 5 ⊢ Ⅎ𝑧(¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥)
5		nfeqf 2381	. . . . . 6 ⊢ ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → Ⅎ𝑤 𝑧 = 𝑥)
6		pm3.21 472	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥 ∧ 𝑤 = 𝑦)))
7	5, 6	spimed 2388	. . . . 5 ⊢ ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → (𝑧 = 𝑥 → ∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)))
8	4, 7	eximd 2209	. . . 4 ⊢ ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → (∃𝑧 𝑧 = 𝑥 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)))
9	1, 8	mpi 20	. . 3 ⊢ ((¬ ∀𝑤 𝑤 = 𝑧 ∧ ¬ ∀𝑤 𝑤 = 𝑥) → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
10	9	ex 413	. 2 ⊢ (¬ ∀𝑤 𝑤 = 𝑧 → (¬ ∀𝑤 𝑤 = 𝑥 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)))
11		ax6e 2383	. . 3 ⊢ ∃𝑧 𝑧 = 𝑦
12		nfae 2433	. . . 4 ⊢ Ⅎ𝑧∀𝑤 𝑤 = 𝑥
13		equvini 2455	. . . . 5 ⊢ (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑤 ∧ 𝑤 = 𝑦))
14		equtrr 2025	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑧 = 𝑤 → 𝑧 = 𝑥))
15	14	anim1d 611	. . . . . 6 ⊢ (𝑤 = 𝑥 → ((𝑧 = 𝑤 ∧ 𝑤 = 𝑦) → (𝑧 = 𝑥 ∧ 𝑤 = 𝑦)))
16	15	aleximi 1834	. . . . 5 ⊢ (∀𝑤 𝑤 = 𝑥 → (∃𝑤(𝑧 = 𝑤 ∧ 𝑤 = 𝑦) → ∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)))
17	13, 16	syl5 34	. . . 4 ⊢ (∀𝑤 𝑤 = 𝑥 → (𝑧 = 𝑦 → ∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)))
18	12, 17	eximd 2209	. . 3 ⊢ (∀𝑤 𝑤 = 𝑥 → (∃𝑧 𝑧 = 𝑦 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)))
19	11, 18	mpi 20	. 2 ⊢ (∀𝑤 𝑤 = 𝑥 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
20	10, 19	pm2.61d2 181	1 ⊢ (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787

This theorem is referenced by:  2ax6e  2471