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Theorem rexn0 3298
Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
Assertion
Ref Expression
rexn0 (x A φA ≠ ∅)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem rexn0
StepHypRef Expression
1 ne0i 3207 . . 3 (x AA ≠ ∅)
21a1d 22 . 2 (x A → (φA ≠ ∅))
32rexlimiv 2405 1 (x A φA ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  wne 2186  wrex 2285  c0 3201
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-nul 3202
This theorem is referenced by: (None)
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