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Theorem dvelimALT 1934
Description: Version of dvelim 1941 that doesn't use ax-10 1441. Because it has different distinct variable constraints than dvelim 1941 and is used in important proofs, it would be better if it had a name which does not end in ALT (ideally more close to set.mm naming). (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvelimALT.1 (𝜑 → ∀𝑥𝜑)
dvelimALT.2 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimALT (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Distinct variable groups:   𝜓,𝑧   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)

Proof of Theorem dvelimALT
StepHypRef Expression
1 nfv 1466 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
2 ax-i12 1443 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
3 orcom 682 . . . . . . . . . 10 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ↔ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
43orbi2i 714 . . . . . . . . 9 ((∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))) ↔ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦)))
52, 4mpbi 143 . . . . . . . 8 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
6 orass 719 . . . . . . . 8 (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦) ↔ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦)))
75, 6mpbir 144 . . . . . . 7 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦)
8 nfa1 1479 . . . . . . . . . . 11 𝑥𝑥 𝑥 = 𝑧
9 ax16ALT 1787 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑧 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
108, 9nfd 1461 . . . . . . . . . 10 (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑧 = 𝑦)
11 dvelimALT.1 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝜑)
1211nfi 1396 . . . . . . . . . . 11 𝑥𝜑
1312a1i 9 . . . . . . . . . 10 (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥𝜑)
1410, 13nfimd 1522 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
15 df-nf 1395 . . . . . . . . . 10 (Ⅎ𝑥 𝑧 = 𝑦 ↔ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
16 id 19 . . . . . . . . . . 11 (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
1712a1i 9 . . . . . . . . . . 11 (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥𝜑)
1816, 17nfimd 1522 . . . . . . . . . 10 (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
1915, 18sylbir 133 . . . . . . . . 9 (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦𝜑))
2014, 19jaoi 671 . . . . . . . 8 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → Ⅎ𝑥(𝑧 = 𝑦𝜑))
2120orim1i 712 . . . . . . 7 (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦) → (Ⅎ𝑥(𝑧 = 𝑦𝜑) ∨ ∀𝑥 𝑥 = 𝑦))
227, 21ax-mp 7 . . . . . 6 (Ⅎ𝑥(𝑧 = 𝑦𝜑) ∨ ∀𝑥 𝑥 = 𝑦)
23 orcom 682 . . . . . 6 ((Ⅎ𝑥(𝑧 = 𝑦𝜑) ∨ ∀𝑥 𝑥 = 𝑦) ↔ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥(𝑧 = 𝑦𝜑)))
2422, 23mpbi 143 . . . . 5 (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥(𝑧 = 𝑦𝜑))
2524ori 677 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
261, 25nfald 1690 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑))
27 ax-17 1464 . . . . 5 (𝜓 → ∀𝑧𝜓)
28 dvelimALT.2 . . . . 5 (𝑧 = 𝑦 → (𝜑𝜓))
2927, 28equsalh 1661 . . . 4 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
3029nfbii 1407 . . 3 (Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑) ↔ Ⅎ𝑥𝜓)
3126, 30sylib 120 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
3231nfrd 1458 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 664  wal 1287  wnf 1394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  hbsb4  1936
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