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Theorem nfabdw 2318
Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2319 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfabdw.1 𝑦𝜑
nfabdw.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfabdw (𝜑𝑥{𝑦𝜓})
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfabdw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1508 . 2 𝑧𝜑
2 df-clab 2144 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabdw.1 . . . . 5 𝑦𝜑
4 nfabdw.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
53, 4alrimi 1502 . . . 4 (𝜑 → ∀𝑦𝑥𝜓)
6 nfa1 1521 . . . . . . . . 9 𝑦𝑦𝑥𝜓
7 sb6 1866 . . . . . . . . . . . 12 ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧𝜓))
87a1i 9 . . . . . . . . . . 11 (∀𝑦𝑥𝜓 → ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧𝜓)))
97biimpri 132 . . . . . . . . . . . 12 (∀𝑦(𝑦 = 𝑧𝜓) → [𝑧 / 𝑦]𝜓)
109axc4i 1522 . . . . . . . . . . 11 (∀𝑦(𝑦 = 𝑧𝜓) → ∀𝑦[𝑧 / 𝑦]𝜓)
118, 10syl6bi 162 . . . . . . . . . 10 (∀𝑦𝑥𝜓 → ([𝑧 / 𝑦]𝜓 → ∀𝑦[𝑧 / 𝑦]𝜓))
126, 11nf5d 2005 . . . . . . . . 9 (∀𝑦𝑥𝜓 → Ⅎ𝑦[𝑧 / 𝑦]𝜓)
136, 12nfim1 1551 . . . . . . . 8 𝑦(∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓)
14 sbequ12 1751 . . . . . . . . 9 (𝑦 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑦]𝜓))
1514imbi2d 229 . . . . . . . 8 (𝑦 = 𝑧 → ((∀𝑦𝑥𝜓𝜓) ↔ (∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓)))
1613, 15equsalv 1773 . . . . . . 7 (∀𝑦(𝑦 = 𝑧 → (∀𝑦𝑥𝜓𝜓)) ↔ (∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓))
1716bicomi 131 . . . . . 6 ((∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ ∀𝑦(𝑦 = 𝑧 → (∀𝑦𝑥𝜓𝜓)))
18 nfv 1508 . . . . . . . 8 𝑥 𝑦 = 𝑧
19 nfnf1 1524 . . . . . . . . . 10 𝑥𝑥𝜓
2019nfal 1556 . . . . . . . . 9 𝑥𝑦𝑥𝜓
21 sp 1491 . . . . . . . . 9 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝜓)
2220, 21nfim1 1551 . . . . . . . 8 𝑥(∀𝑦𝑥𝜓𝜓)
2318, 22nfim 1552 . . . . . . 7 𝑥(𝑦 = 𝑧 → (∀𝑦𝑥𝜓𝜓))
2423nfal 1556 . . . . . 6 𝑥𝑦(𝑦 = 𝑧 → (∀𝑦𝑥𝜓𝜓))
2517, 24nfxfr 1454 . . . . 5 𝑥(∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓)
26 pm5.5 241 . . . . . 6 (∀𝑦𝑥𝜓 → ((∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ [𝑧 / 𝑦]𝜓))
2720, 26nfbidf 1519 . . . . 5 (∀𝑦𝑥𝜓 → (Ⅎ𝑥(∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜓))
2825, 27mpbii 147 . . . 4 (∀𝑦𝑥𝜓 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
295, 28syl 14 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
302, 29nfxfrd 1455 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
311, 30nfcd 2294 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1333  wnf 1440  [wsb 1742  wcel 2128  {cab 2143  wnfc 2286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-nfc 2288
This theorem is referenced by:  nfsbcdw  3065  nfcsbw  3067
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