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Theorem nfreuxy 2501
Description: Not-free for restricted uniqueness. This is a version where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
Hypotheses
Ref Expression
nfreuxy.1 𝑥𝐴
nfreuxy.2 𝑥𝜑
Assertion
Ref Expression
nfreuxy 𝑥∃!𝑦𝐴 𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfreuxy
StepHypRef Expression
1 nftru 1371 . . 3 𝑦
2 nfreuxy.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfreuxy.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfreudxy 2500 . 2 (⊤ → Ⅎ𝑥∃!𝑦𝐴 𝜑)
76trud 1268 1 𝑥∃!𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1260  wnf 1365  wnfc 2181  ∃!wreu 2325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-cleq 2049  df-clel 2052  df-nfc 2183  df-reu 2330
This theorem is referenced by:  sbcreug  2865
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