Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfrexdxy GIF version

Theorem nfrexdxy 2491
 Description: Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexdya 2493 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfraldxy.2 𝑦𝜑
nfraldxy.3 (𝜑𝑥𝐴)
nfraldxy.4 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfrexdxy (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfrexdxy
StepHypRef Expression
1 df-rex 2441 . 2 (∃𝑦𝐴 𝜓 ↔ ∃𝑦(𝑦𝐴𝜓))
2 nfraldxy.2 . . 3 𝑦𝜑
3 nfcv 2299 . . . . . 6 𝑥𝑦
43a1i 9 . . . . 5 (𝜑𝑥𝑦)
5 nfraldxy.3 . . . . 5 (𝜑𝑥𝐴)
64, 5nfeld 2315 . . . 4 (𝜑 → Ⅎ𝑥 𝑦𝐴)
7 nfraldxy.4 . . . 4 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfand 1548 . . 3 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
92, 8nfexd 1741 . 2 (𝜑 → Ⅎ𝑥𝑦(𝑦𝐴𝜓))
101, 9nfxfrd 1455 1 (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  Ⅎwnf 1440  ∃wex 1472   ∈ wcel 2128  Ⅎwnfc 2286  ∃wrex 2436 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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441 This theorem is referenced by:  nfrexdya  2493  nfrexxy  2496  nfunid  3779  strcollnft  13519
 Copyright terms: Public domain W3C validator