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Theorem nfabdw 2338
Description: Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2339 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 1528 . 2 𝑧𝜑
2 df-clab 2164 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
3 nfabdw.1 . . . . 5 𝑦𝜑
4 nfabdw.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
53, 4alrimi 1522 . . . 4 (𝜑 → ∀𝑦𝑥𝜓)
6 nfa1 1541 . . . . . . . . 9 𝑦𝑦𝑥𝜓
7 sb6 1886 . . . . . . . . . . . 12 ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧𝜓))
87a1i 9 . . . . . . . . . . 11 (∀𝑦𝑥𝜓 → ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧𝜓)))
97biimpri 133 . . . . . . . . . . . 12 (∀𝑦(𝑦 = 𝑧𝜓) → [𝑧 / 𝑦]𝜓)
109axc4i 1542 . . . . . . . . . . 11 (∀𝑦(𝑦 = 𝑧𝜓) → ∀𝑦[𝑧 / 𝑦]𝜓)
118, 10syl6bi 163 . . . . . . . . . 10 (∀𝑦𝑥𝜓 → ([𝑧 / 𝑦]𝜓 → ∀𝑦[𝑧 / 𝑦]𝜓))
126, 11nf5d 2025 . . . . . . . . 9 (∀𝑦𝑥𝜓 → Ⅎ𝑦[𝑧 / 𝑦]𝜓)
136, 12nfim1 1571 . . . . . . . 8 𝑦(∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓)
14 sbequ12 1771 . . . . . . . . 9 (𝑦 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑦]𝜓))
1514imbi2d 230 . . . . . . . 8 (𝑦 = 𝑧 → ((∀𝑦𝑥𝜓𝜓) ↔ (∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓)))
1613, 15equsalv 1793 . . . . . . 7 (∀𝑦(𝑦 = 𝑧 → (∀𝑦𝑥𝜓𝜓)) ↔ (∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓))
1716bicomi 132 . . . . . 6 ((∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ ∀𝑦(𝑦 = 𝑧 → (∀𝑦𝑥𝜓𝜓)))
18 nfv 1528 . . . . . . . 8 𝑥 𝑦 = 𝑧
19 nfnf1 1544 . . . . . . . . . 10 𝑥𝑥𝜓
2019nfal 1576 . . . . . . . . 9 𝑥𝑦𝑥𝜓
21 sp 1511 . . . . . . . . 9 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝜓)
2220, 21nfim1 1571 . . . . . . . 8 𝑥(∀𝑦𝑥𝜓𝜓)
2318, 22nfim 1572 . . . . . . 7 𝑥(𝑦 = 𝑧 → (∀𝑦𝑥𝜓𝜓))
2423nfal 1576 . . . . . 6 𝑥𝑦(𝑦 = 𝑧 → (∀𝑦𝑥𝜓𝜓))
2517, 24nfxfr 1474 . . . . 5 𝑥(∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓)
26 pm5.5 242 . . . . . 6 (∀𝑦𝑥𝜓 → ((∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ [𝑧 / 𝑦]𝜓))
2720, 26nfbidf 1539 . . . . 5 (∀𝑦𝑥𝜓 → (Ⅎ𝑥(∀𝑦𝑥𝜓 → [𝑧 / 𝑦]𝜓) ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜓))
2825, 27mpbii 148 . . . 4 (∀𝑦𝑥𝜓 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
295, 28syl 14 . . 3 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
302, 29nfxfrd 1475 . 2 (𝜑 → Ⅎ𝑥 𝑧 ∈ {𝑦𝜓})
311, 30nfcd 2314 1 (𝜑𝑥{𝑦𝜓})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351  wnf 1460  [wsb 1762  wcel 2148  {cab 2163  wnfc 2306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-nfc 2308
This theorem is referenced by:  nfsbcdw  3091  nfcsbw  3093
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