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Theorem rexnalim 2295
Description: Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
rexnalim (x A ¬ φ → ¬ x A φ)

Proof of Theorem rexnalim
StepHypRef Expression
1 df-rex 2290 . 2 (x A ¬ φx(x A ¬ φ))
2 exanaliim 1520 . . 3 (x(x A ¬ φ) → ¬ x(x Aφ))
3 df-ral 2289 . . 3 (x A φx(x Aφ))
42, 3sylnibr 589 . 2 (x(x A ¬ φ) → ¬ x A φ)
51, 4sylbi 114 1 (x A ¬ φ → ¬ x A φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wal 1226  wex 1362   wcel 1374  wral 2284  wrex 2285
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-in1 532  ax-in2 533  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-ral 2289  df-rex 2290
This theorem is referenced by:  ralexim  2296  iundif2ss  3696
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