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Theorem nfrexdya 2526
Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexdxy 2524 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 2474 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
2 sban 1967 . . . . . 6 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦]𝜓))
3 clelsb1 2294 . . . . . . 7 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43anbi1i 458 . . . . . 6 (([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦]𝜓) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
52, 4bitri 184 . . . . 5 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
65exbii 1616 . . . 4 (∃𝑧[𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
7 nfv 1539 . . . . 5 𝑧(𝑦𝐴𝜓)
87sb8e 1868 . . . 4 (∃𝑦(𝑦𝐴𝜓) ↔ ∃𝑧[𝑧 / 𝑦](𝑦𝐴𝜓))
9 df-rex 2474 . . . 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, 15nfrexdxy 2524 . . 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  wa 104  wnf 1471  wex 1503  [wsb 1773  wcel 2160  wnfc 2319  wrex 2469
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-rex 2474
This theorem is referenced by:  nfrexya  2531
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