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Theorem ax6e2ndeqALT 42038
 Description: "At least two sets exist" expressed in the form of dtru 5242 is logically equivalent to the same expressed in a form similar to ax6e 2390 if dtru 5242 is false implies 𝑢 = 𝑣. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in ax6e2ndeqVD 42016. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ax6e2ndeqALT ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Distinct variable groups:   𝑥,𝑢   𝑦,𝑢   𝑥,𝑣   𝑦,𝑣

Proof of Theorem ax6e2ndeqALT
StepHypRef Expression
1 ax6e2nd 41665 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 ax6e2eq 41664 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
31a1d 25 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
4 exmid 892 . . . 4 (∀𝑥 𝑥 = 𝑦 ∨ ¬ ∀𝑥 𝑥 = 𝑦)
5 jao 958 . . . . 5 ((∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))) → ((¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))) → ((∀𝑥 𝑥 = 𝑦 ∨ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))))
653imp 1108 . . . 4 (((∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))) ∧ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))) ∧ (∀𝑥 𝑥 = 𝑦 ∨ ¬ ∀𝑥 𝑥 = 𝑦)) → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
72, 3, 4, 6mp3an 1458 . . 3 (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
81, 7jaoi 854 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
9 hbnae 2443 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
109eximi 1836 . . . . . . . 8 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
11 nfa1 2152 . . . . . . . . 9 𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦
121119.9 2203 . . . . . . . 8 (∃𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
1310, 12sylib 221 . . . . . . 7 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
14 sp 2180 . . . . . . 7 (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
1513, 14syl 17 . . . . . 6 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
16 excom 2166 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
17 nfa1 2152 . . . . . . . . . . . 12 𝑥𝑥 𝑥 = 𝑦
1817nfn 1858 . . . . . . . . . . 11 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
191819.9 2203 . . . . . . . . . 10 (∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
20 id 22 . . . . . . . . . . . . . . . 16 (𝑢𝑣𝑢𝑣)
21 simpr 488 . . . . . . . . . . . . . . . . 17 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → (𝑥 = 𝑢𝑦 = 𝑣))
22 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
2321, 22syl 17 . . . . . . . . . . . . . . . 16 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → 𝑥 = 𝑢)
24 pm13.181 3033 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑢𝑢𝑣) → 𝑥𝑣)
2524ancoms 462 . . . . . . . . . . . . . . . 16 ((𝑢𝑣𝑥 = 𝑢) → 𝑥𝑣)
2620, 23, 25syl2an2r 684 . . . . . . . . . . . . . . 15 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → 𝑥𝑣)
27 simpr 488 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
2821, 27syl 17 . . . . . . . . . . . . . . 15 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → 𝑦 = 𝑣)
29 neeq2 3014 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 → (𝑥𝑦𝑥𝑣))
3029biimparc 483 . . . . . . . . . . . . . . 15 ((𝑥𝑣𝑦 = 𝑣) → 𝑥𝑦)
3126, 28, 30syl2anc 587 . . . . . . . . . . . . . 14 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → 𝑥𝑦)
32 df-ne 2952 . . . . . . . . . . . . . . . 16 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
3332bicomi 227 . . . . . . . . . . . . . . 15 𝑥 = 𝑦𝑥𝑦)
34 sp 2180 . . . . . . . . . . . . . . . 16 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
3534con3i 157 . . . . . . . . . . . . . . 15 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3633, 35sylbir 238 . . . . . . . . . . . . . 14 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3731, 36syl 17 . . . . . . . . . . . . 13 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → ¬ ∀𝑥 𝑥 = 𝑦)
3837ex 416 . . . . . . . . . . . 12 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
3938alrimiv 1928 . . . . . . . . . . 11 (𝑢𝑣 → ∀𝑥((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
40 exim 1835 . . . . . . . . . . 11 (∀𝑥((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦))
4139, 40syl 17 . . . . . . . . . 10 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦))
42 imbi2 352 . . . . . . . . . . 11 ((∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) → ((∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦) ↔ (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)))
4342biimpa 480 . . . . . . . . . 10 (((∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) ∧ (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦)) → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
4419, 41, 43sylancr 590 . . . . . . . . 9 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
4544alrimiv 1928 . . . . . . . 8 (𝑢𝑣 → ∀𝑦(∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
46 exim 1835 . . . . . . . 8 (∀𝑦(∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
4745, 46syl 17 . . . . . . 7 (𝑢𝑣 → (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
48 imbi1 351 . . . . . . . 8 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)) → ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦)))
4948biimpar 481 . . . . . . 7 (((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣)) ∧ (∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦)) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
5016, 47, 49sylancr 590 . . . . . 6 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
51 pm3.34 765 . . . . . 6 (((∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦) ∧ (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦)) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
5215, 50, 51sylancr 590 . . . . 5 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
53 orc 864 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
5453imim2i 16 . . . . 5 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
5552, 54syl 17 . . . 4 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
5655idiALT 41584 . . 3 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
57 id 22 . . . . . 6 (𝑢 = 𝑣𝑢 = 𝑣)
58 ax-1 6 . . . . . 6 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → 𝑢 = 𝑣))
5957, 58syl 17 . . . . 5 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → 𝑢 = 𝑣))
60 olc 865 . . . . . 6 (𝑢 = 𝑣 → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
6160imim2i 16 . . . . 5 ((∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → 𝑢 = 𝑣) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
6259, 61syl 17 . . . 4 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
6362idiALT 41584 . . 3 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
64 exmidne 2961 . . 3 (𝑢 = 𝑣𝑢𝑣)
65 jao 958 . . . 4 ((𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))) → ((𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))) → ((𝑢 = 𝑣𝑢𝑣) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))))
66653imp21 1111 . . 3 (((𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))) ∧ (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))) ∧ (𝑢 = 𝑣𝑢𝑣)) → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
6756, 63, 64, 66mp3an 1458 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
688, 67impbii 212 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538  ∃wex 1781   ≠ wne 2951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411 This theorem is referenced by: (None)
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