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Theorem nfrexdya 2402
Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexdxy 2400 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
nfrexdya (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfrexdya
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-rex 2355 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
2 sban 1871 . . . . . 6 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦]𝜓))
3 clelsb3 2184 . . . . . . 7 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43anbi1i 446 . . . . . 6 (([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦]𝜓) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
52, 4bitri 182 . . . . 5 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
65exbii 1537 . . . 4 (∃𝑧[𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
7 nfv 1462 . . . . 5 𝑧(𝑦𝐴𝜓)
87sb8e 1779 . . . 4 (∃𝑦(𝑦𝐴𝜓) ↔ ∃𝑧[𝑧 / 𝑦](𝑦𝐴𝜓))
9 df-rex 2355 . . . 4 (∃𝑧𝐴 [𝑧 / 𝑦]𝜓 ↔ ∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
106, 8, 93bitr4i 210 . . 3 (∃𝑦(𝑦𝐴𝜓) ↔ ∃𝑧𝐴 [𝑧 / 𝑦]𝜓)
11 nfv 1462 . . . 4 𝑧𝜑
12 nfraldya.3 . . . 4 (𝜑𝑥𝐴)
13 nfraldya.2 . . . . 5 𝑦𝜑
14 nfraldya.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
1513, 14nfsbd 1893 . . . 4 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
1611, 12, 15nfrexdxy 2400 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 [𝑧 / 𝑦]𝜓)
1710, 16nfxfrd 1405 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
181, 17nfxfrd 1405 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wnf 1390  wex 1422  wcel 1434  [wsb 1686  wnfc 2207  wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355
This theorem is referenced by:  nfrexya  2406
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