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Theorem dvelimfv 1903
Description: Like dvelimf 1907 but with a distinct variable constraint on 𝑥 and 𝑧. (Contributed by Jim Kingdon, 6-Mar-2018.)
Hypotheses
Ref Expression
dvelimfv.1 (𝜑 → ∀𝑥𝜑)
dvelimfv.2 (𝜓 → ∀𝑧𝜓)
dvelimfv.3 (𝑧 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
dvelimfv (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem dvelimfv
StepHypRef Expression
1 nfnae 1626 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
2 ax-i12 1414 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)))
3 orcom 657 . . . . . . . . . 10 ((∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ↔ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
43orbi2i 689 . . . . . . . . 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 137 . . . . . . . 8 (∀𝑥 𝑥 = 𝑧 ∨ (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) ∨ ∀𝑥 𝑥 = 𝑦))
6 orass 694 . . . . . . . 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 138 . . . . . . 7 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ∨ ∀𝑥 𝑥 = 𝑦)
8 nfae 1623 . . . . . . . . . . 11 𝑥𝑥 𝑥 = 𝑧
9 ax16ALT 1755 . . . . . . . . . . 11 (∀𝑥 𝑥 = 𝑧 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
108, 9nfd 1432 . . . . . . . . . 10 (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑧 = 𝑦)
11 dvelimfv.1 . . . . . . . . . . . 12 (𝜑 → ∀𝑥𝜑)
1211nfi 1367 . . . . . . . . . . 11 𝑥𝜑
1312a1i 9 . . . . . . . . . 10 (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥𝜑)
1410, 13nfimd 1493 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
15 df-nf 1366 . . . . . . . . . 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 1493 . . . . . . . . . 10 (Ⅎ𝑥 𝑧 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
1915, 18sylbir 129 . . . . . . . . 9 (∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦) → Ⅎ𝑥(𝑧 = 𝑦𝜑))
2014, 19jaoi 646 . . . . . . . 8 ((∀𝑥 𝑥 = 𝑧 ∨ ∀𝑥(𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) → Ⅎ𝑥(𝑧 = 𝑦𝜑))
2120orim1i 687 . . . . . . 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 657 . . . . . 6 ((Ⅎ𝑥(𝑧 = 𝑦𝜑) ∨ ∀𝑥 𝑥 = 𝑦) ↔ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥(𝑧 = 𝑦𝜑)))
2422, 23mpbi 137 . . . . 5 (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥(𝑧 = 𝑦𝜑))
2524ori 652 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦𝜑))
261, 25nfald 1659 . . 3 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑))
27 dvelimfv.2 . . . . 5 (𝜓 → ∀𝑧𝜓)
28 dvelimfv.3 . . . . 5 (𝑧 = 𝑦 → (𝜑𝜓))
2927, 28equsalh 1630 . . . 4 (∀𝑧(𝑧 = 𝑦𝜑) ↔ 𝜓)
3029nfbii 1378 . . 3 (Ⅎ𝑥𝑧(𝑧 = 𝑦𝜑) ↔ Ⅎ𝑥𝜓)
3126, 30sylib 131 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓)
3231nfrd 1429 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  wal 1257  wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662
This theorem is referenced by: (None)
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