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Theorem nfralxy 2338
Description: Not-free for restricted universal quantification where x and y are distinct. See nfralya 2340 for a version with y and A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
Hypotheses
Ref Expression
nfralxy.1 xA
nfralxy.2 xφ
Assertion
Ref Expression
nfralxy xy A φ
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   A(x,y)

Proof of Theorem nfralxy
StepHypRef Expression
1 nftru 1335 . . 3 y
2 nfralxy.1 . . . 4 xA
32a1i 9 . . 3 ( ⊤ → xA)
4 nfralxy.2 . . . 4 xφ
54a1i 9 . . 3 ( ⊤ → Ⅎxφ)
61, 3, 5nfraldxy 2334 . 2 ( ⊤ → Ⅎxy A φ)
76trud 1237 1 xy A φ
Colors of variables: wff set class
Syntax hints:  wtru 1229  wnf 1329  wnfc 2147  wral 2284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289
This theorem is referenced by:  nfra2xy  2342  rspc2  2638  sbcralt  2811  sbcralg  2813  raaanlem  3305  nfint  3599  nfiinxy  3658  nfpo  4012  nfso  4013  nfse  4046  ralxpf  4409  funimaexglem  4908  fun11iun  5072  dff13f  5334  nfiso  5371  mpt2eq123  5487  fmpt2x  5749  nfrecs  5844  setindis  7332  bdsetindis  7334
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