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Theorem nfrexdya 2506
Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexdxy 2504 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 2454 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
2 sban 1948 . . . . . 6 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦]𝜓))
3 clelsb1 2275 . . . . . . 7 ([𝑧 / 𝑦]𝑦𝐴𝑧𝐴)
43anbi1i 455 . . . . . 6 (([𝑧 / 𝑦]𝑦𝐴 ∧ [𝑧 / 𝑦]𝜓) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
52, 4bitri 183 . . . . 5 ([𝑧 / 𝑦](𝑦𝐴𝜓) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
65exbii 1598 . . . 4 (∃𝑧[𝑧 / 𝑦](𝑦𝐴𝜓) ↔ ∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
7 nfv 1521 . . . . 5 𝑧(𝑦𝐴𝜓)
87sb8e 1850 . . . 4 (∃𝑦(𝑦𝐴𝜓) ↔ ∃𝑧[𝑧 / 𝑦](𝑦𝐴𝜓))
9 df-rex 2454 . . . 4 (∃𝑧𝐴 [𝑧 / 𝑦]𝜓 ↔ ∃𝑧(𝑧𝐴 ∧ [𝑧 / 𝑦]𝜓))
106, 8, 93bitr4i 211 . . 3 (∃𝑦(𝑦𝐴𝜓) ↔ ∃𝑧𝐴 [𝑧 / 𝑦]𝜓)
11 nfv 1521 . . . 4 𝑧𝜑
12 nfraldya.3 . . . 4 (𝜑𝑥𝐴)
13 nfraldya.2 . . . . 5 𝑦𝜑
14 nfraldya.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
1513, 14nfsbd 1970 . . . 4 (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
1611, 12, 15nfrexdxy 2504 . . 3 (𝜑 → Ⅎ𝑥𝑧𝐴 [𝑧 / 𝑦]𝜓)
1710, 16nfxfrd 1468 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
181, 17nfxfrd 1468 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wnf 1453  wex 1485  [wsb 1755  wcel 2141  wnfc 2299  wrex 2449
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454
This theorem is referenced by:  nfrexya  2511
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