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Theorem ax6e2ndeqALT 41149
Description: "At least two sets exist" expressed in the form of dtru 5268 is logically equivalent to the same expressed in a form similar to ax6e 2396 if dtru 5268 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 41127. (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 40776 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
2 ax6e2eq 40775 . . . 4 (∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
31a1d 25 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣)))
4 exmid 890 . . . 4 (∀𝑥 𝑥 = 𝑦 ∨ ¬ ∀𝑥 𝑥 = 𝑦)
5 jao 956 . . . . 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 1105 . . . 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 1454 . . 3 (𝑢 = 𝑣 → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
81, 7jaoi 853 . 2 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) → ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
9 hbnae 2451 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
109eximi 1828 . . . . . . . 8 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∃𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
11 nfa1 2148 . . . . . . . . 9 𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦
121119.9 2198 . . . . . . . 8 (∃𝑦𝑦 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
1310, 12sylib 219 . . . . . . 7 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦)
14 sp 2174 . . . . . . 7 (∀𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
1513, 14syl 17 . . . . . 6 (∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
16 excom 2161 . . . . . . 7 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) ↔ ∃𝑦𝑥(𝑥 = 𝑢𝑦 = 𝑣))
17 nfa1 2148 . . . . . . . . . . . 12 𝑥𝑥 𝑥 = 𝑦
1817nfn 1850 . . . . . . . . . . 11 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
191819.9 2198 . . . . . . . . . 10 (∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
20 id 22 . . . . . . . . . . . . . . . 16 (𝑢𝑣𝑢𝑣)
21 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → (𝑥 = 𝑢𝑦 = 𝑣))
22 simpl 483 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑥 = 𝑢)
2321, 22syl 17 . . . . . . . . . . . . . . . 16 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → 𝑥 = 𝑢)
24 pm13.181 3104 . . . . . . . . . . . . . . . . 17 ((𝑥 = 𝑢𝑢𝑣) → 𝑥𝑣)
2524ancoms 459 . . . . . . . . . . . . . . . 16 ((𝑢𝑣𝑥 = 𝑢) → 𝑥𝑣)
2620, 23, 25syl2an2r 681 . . . . . . . . . . . . . . 15 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → 𝑥𝑣)
27 simpr 485 . . . . . . . . . . . . . . . 16 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝑦 = 𝑣)
2821, 27syl 17 . . . . . . . . . . . . . . 15 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → 𝑦 = 𝑣)
29 neeq2 3084 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑣 → (𝑥𝑦𝑥𝑣))
3029biimparc 480 . . . . . . . . . . . . . . 15 ((𝑥𝑣𝑦 = 𝑣) → 𝑥𝑦)
3126, 28, 30syl2anc 584 . . . . . . . . . . . . . 14 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → 𝑥𝑦)
32 df-ne 3022 . . . . . . . . . . . . . . . 16 (𝑥𝑦 ↔ ¬ 𝑥 = 𝑦)
3332bicomi 225 . . . . . . . . . . . . . . 15 𝑥 = 𝑦𝑥𝑦)
34 sp 2174 . . . . . . . . . . . . . . . 16 (∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)
3534con3i 157 . . . . . . . . . . . . . . 15 𝑥 = 𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3633, 35sylbir 236 . . . . . . . . . . . . . 14 (𝑥𝑦 → ¬ ∀𝑥 𝑥 = 𝑦)
3731, 36syl 17 . . . . . . . . . . . . 13 ((𝑢𝑣 ∧ (𝑥 = 𝑢𝑦 = 𝑣)) → ¬ ∀𝑥 𝑥 = 𝑦)
3837ex 413 . . . . . . . . . . . 12 (𝑢𝑣 → ((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
3938alrimiv 1921 . . . . . . . . . . 11 (𝑢𝑣 → ∀𝑥((𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
40 exim 1827 . . . . . . . . . . 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 350 . . . . . . . . . . 11 ((∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) → ((∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑥 ¬ ∀𝑥 𝑥 = 𝑦) ↔ (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦)))
4342biimpa 477 . . . . . . . . . 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 587 . . . . . . . . 9 (𝑢𝑣 → (∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
4544alrimiv 1921 . . . . . . . 8 (𝑢𝑣 → ∀𝑦(∃𝑥(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
46 exim 1827 . . . . . . . 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 349 . . . . . . . 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 478 . . . . . . 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 587 . . . . . 6 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ∃𝑦 ¬ ∀𝑥 𝑥 = 𝑦))
51 pm3.34 762 . . . . . 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 587 . . . . 5 (𝑢𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → ¬ ∀𝑥 𝑥 = 𝑦))
53 orc 863 . . . . . 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 40695 . . 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 864 . . . . . 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 40695 . . 3 (𝑢 = 𝑣 → (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣)))
64 exmidne 3031 . . 3 (𝑢 = 𝑣𝑢𝑣)
65 jao 956 . . . 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 1108 . . 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 1454 . 2 (∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣) → (¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣))
688, 67impbii 210 1 ((¬ ∀𝑥 𝑥 = 𝑦𝑢 = 𝑣) ↔ ∃𝑥𝑦(𝑥 = 𝑢𝑦 = 𝑣))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843  wal 1528   = wceq 1530  wex 1773  wne 3021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-ne 3022  df-v 3502
This theorem is referenced by: (None)
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