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Theorem nfraldya 2525
Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfraldxy 2523 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfraldya.2 𝑦𝜑
nfraldya.3 (𝜑𝑥𝐴)
nfraldya.4 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfraldya (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfraldya
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ral 2473 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 sbim 1965 . . . . . 6 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ([𝑧 / 𝑦]𝑦𝐴 → [𝑧 / 𝑦]𝜓))
3 clelsb1 2294 . . . . . . 7 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43imbi1i 238 . . . . . 6 (([𝑧 / 𝑦]𝑦𝐴 → [𝑧 / 𝑦]𝜓) ↔ (𝑧𝐴 → [𝑧 / 𝑦]𝜓))
52, 4bitri 184 . . . . 5 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ (𝑧𝐴 → [𝑧 / 𝑦]𝜓))
65albii 1481 . . . 4 (∀𝑧[𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑦]𝜓))
7 nfv 1539 . . . . 5 𝑧(𝑦𝐴𝜓)
87sb8 1867 . . . 4 (∀𝑦(𝑦𝐴𝜓) ↔ ∀𝑧[𝑧 / 𝑦](𝑦𝐴𝜓))
9 df-ral 2473 . . . 4 (∀𝑧𝐴 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑦]𝜓))
106, 8, 93bitr4i 212 . . 3 (∀𝑦(𝑦𝐴𝜓) ↔ ∀𝑧𝐴 [𝑧 / 𝑦]𝜓)
11 nfv 1539 . . . 4 𝑧𝜑
12 nfraldya.3 . . . 4 (𝜑𝑥𝐴)
13 nfraldya.2 . . . . 5 𝑦𝜑
14 nfraldya.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
1513, 14nfsbd 1989 . . . 4 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
1611, 12, 15nfraldxy 2523 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 [𝑧 / 𝑦]𝜓)
1710, 16nfxfrd 1486 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
181, 17nfxfrd 1486 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  wnf 1471  [wsb 1773  wcel 2160  wnfc 2319  wral 2468
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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473
This theorem is referenced by:  nfralya  2530
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