Theorem ax6e2nd 45247

Description: If at least two sets exist (dtru 5420), then the same is true expressed in an alternate form similar to the form of ax6e 2415. ax6e2nd 45247 is derived from ax6e2ndVD 45596. (Contributed by Alan Sare, 25-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax6e2nd	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣

Proof of Theorem ax6e2nd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3459	. . . . . . 7 ⊢ 𝑢 ∈ V
2		ax6e 2415	. . . . . . 7 ⊢ ∃𝑦 𝑦 = 𝑣
3	1, 2	pm3.2i 475	. . . . . 6 ⊢ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣)
4		19.42v 1983	. . . . . . 7 ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣))
5	4	biimpri 231	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ ∃𝑦 𝑦 = 𝑣) → ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣))
6	3, 5	ax-mp 5	. . . . 5 ⊢ ∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣)
7		isset 3469	. . . . . . 7 ⊢ (𝑢 ∈ V ↔ ∃𝑥 𝑥 = 𝑢)
8	7	anbi1i 635	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ (∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
9	8	exbii 1878	. . . . 5 ⊢ (∃𝑦(𝑢 ∈ V ∧ 𝑦 = 𝑣) ↔ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
10	6, 9	mpbi 233	. . . 4 ⊢ ∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣)
11		id 23	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
12		hbnae 2464	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
13		hbn1 2177	. . . . . . . . . . . 12 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦)
14		ax-5 1940	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑣 → ∀𝑥 𝑧 = 𝑣)
15		ax-5 1940	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑣 → ∀𝑧 𝑦 = 𝑣)
16		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
17		equequ1 2055	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
18	16, 17	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
19	18	idiALT 45167	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑣 ↔ 𝑦 = 𝑣))
20	14, 15, 19	dvelimh 2482	. . . . . . . . . . . . . . 15 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
21	11, 20	syl 18	. . . . . . . . . . . . . 14 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
22	21	idiALT 45167	. . . . . . . . . . . . 13 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
23	22	alimi 1841	. . . . . . . . . . . 12 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
24	13, 23	syl 18	. . . . . . . . . . 11 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
25	11, 24	syl 18	. . . . . . . . . 10 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣))
26		19.41rg 45239	. . . . . . . . . 10 ⊢ (∀𝑥(𝑦 = 𝑣 → ∀𝑥 𝑦 = 𝑣) → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
27	25, 26	syl 18	. . . . . . . . 9 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
28	27	idiALT 45167	. . . . . . . 8 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
29	28	alimi 1841	. . . . . . 7 ⊢ (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
30	12, 29	syl 18	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
31	11, 30	syl 18	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
32		exim 1864	. . . . 5 ⊢ (∀𝑦((∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) → (∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
33	31, 32	syl 18	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
34		pm2.27 43	. . . 4 ⊢ (∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((∃𝑦(∃𝑥 𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)))
35	10, 33, 34	mpsyl 69	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
36		excomim 2198	. . 3 ⊢ (∃𝑦∃𝑥(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
37	35, 36	syl 18	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))
38	37	idiALT 45167	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  ∃wex 1809   ∈ wcel 2143  Vcvv 3455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457

This theorem is referenced by:  ax6e2ndeq  45248  ax6e2ndeqVD  45597  ax6e2ndeqALT  45619