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Theorem nfraldya 2469
Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfraldxy 2467 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 2421 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2 sbim 1926 . . . . . 6 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ([𝑧 / 𝑦]𝑦𝐴 → [𝑧 / 𝑦]𝜓))
3 clelsb3 2244 . . . . . . 7 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43imbi1i 237 . . . . . 6 (([𝑧 / 𝑦]𝑦𝐴 → [𝑧 / 𝑦]𝜓) ↔ (𝑧𝐴 → [𝑧 / 𝑦]𝜓))
52, 4bitri 183 . . . . 5 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ (𝑧𝐴 → [𝑧 / 𝑦]𝜓))
65albii 1446 . . . 4 (∀𝑧[𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑦]𝜓))
7 nfv 1508 . . . . 5 𝑧(𝑦𝐴𝜓)
87sb8 1828 . . . 4 (∀𝑦(𝑦𝐴𝜓) ↔ ∀𝑧[𝑧 / 𝑦](𝑦𝐴𝜓))
9 df-ral 2421 . . . 4 (∀𝑧𝐴 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑦]𝜓))
106, 8, 93bitr4i 211 . . 3 (∀𝑦(𝑦𝐴𝜓) ↔ ∀𝑧𝐴 [𝑧 / 𝑦]𝜓)
11 nfv 1508 . . . 4 𝑧𝜑
12 nfraldya.3 . . . 4 (𝜑𝑥𝐴)
13 nfraldya.2 . . . . 5 𝑦𝜑
14 nfraldya.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
1513, 14nfsbd 1950 . . . 4 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
1611, 12, 15nfraldxy 2467 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 [𝑧 / 𝑦]𝜓)
1710, 16nfxfrd 1451 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
181, 17nfxfrd 1451 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1329  wnf 1436  wcel 1480  [wsb 1735  wnfc 2268  wral 2416
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421
This theorem is referenced by:  nfralya  2473
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