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Theorem dvelimor 1942
Description: Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula 𝜑 (containing 𝑧) and a distinct variable constraint between 𝑥 and 𝑧. The theorem makes it possible to replace the distinct variable constraint with the disjunct 𝑥𝑥 = 𝑦 (𝜓 is just a version of 𝜑 with 𝑦 substituted for 𝑧). (Contributed by Jim Kingdon, 11-May-2018.)
Hypotheses
Ref Expression
dvelimor.1 𝑥𝜑
dvelimor.2 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimor (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥𝜓)
Distinct variable groups:   𝜓,𝑧   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦)

Proof of Theorem dvelimor
StepHypRef Expression
1 ax-bndl 1444 . . . . . 6 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
2 orcom 682 . . . . . . 7 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ↔ (∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
32orbi2i 714 . . . . . 6 ((∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))) ↔ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦)))
41, 3mpbi 143 . . . . 5 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
5 orass 719 . . . . 5 (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦) ↔ (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦)))
64, 5mpbir 144 . . . 4 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦)
7 nfae 1654 . . . . . . 7 𝑧𝑥 𝑥 = 𝑧
8 a16nf 1794 . . . . . . 7 (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
97, 8alrimi 1460 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ∀𝑧𝑥(𝑧 = 𝑦𝜑))
10 df-nf 1395 . . . . . . . 8 (Ⅎ𝑥 𝑧 = 𝑦 ↔ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
11 id 19 . . . . . . . . 9 (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
12 dvelimor.1 . . . . . . . . . 10 𝑥𝜑
1312a1i 9 . . . . . . . . 9 (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥𝜑)
1411, 13nfimd 1522 . . . . . . . 8 (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
1510, 14sylbir 133 . . . . . . 7 (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦𝜑))
1615alimi 1389 . . . . . 6 (∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → ∀𝑧𝑥(𝑧 = 𝑦𝜑))
179, 16jaoi 671 . . . . 5 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → ∀𝑧𝑥(𝑧 = 𝑦𝜑))
1817orim1i 712 . . . 4 (((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑧𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦) → (∀𝑧𝑥(𝑧 = 𝑦𝜑) ∨ ∀𝑥 𝑥 = 𝑦))
196, 18ax-mp 7 . . 3 (∀𝑧𝑥(𝑧 = 𝑦𝜑) ∨ ∀𝑥 𝑥 = 𝑦)
20 orcom 682 . . 3 ((∀𝑧𝑥(𝑧 = 𝑦𝜑) ∨ ∀𝑥 𝑥 = 𝑦) ↔ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧𝑥(𝑧 = 𝑦𝜑)))
2119, 20mpbi 143 . 2 (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧𝑥(𝑧 = 𝑦𝜑))
22 nfalt 1515 . . . 4 (∀𝑧𝑥(𝑧 = 𝑦𝜑) → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑))
23 ax-17 1464 . . . . . 6 (𝜓 → ∀𝑧𝜓)
24 dvelimor.2 . . . . . 6 (𝑧 = 𝑦 → (𝜑𝜓))
2523, 24equsalh 1661 . . . . 5 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
2625nfbii 1407 . . . 4 (Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑) ↔ Ⅎ𝑥𝜓)
2722, 26sylib 120 . . 3 (∀𝑧𝑥(𝑧 = 𝑦𝜑) → Ⅎ𝑥𝜓)
2827orim2i 713 . 2 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑧𝑥(𝑧 = 𝑦𝜑)) → (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥𝜓))
2921, 28ax-mp 7 1 (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  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:  nfsb4or  1947  rgen2a  2429
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