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Theorem nfcrii 2325
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfcri.1  |-  F/_ x A
Assertion
Ref Expression
nfcrii  |-  ( y  e.  A  ->  A. x  y  e.  A )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem nfcrii
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfcri.1 . . . 4  |-  F/_ x A
2 nfcr 2324 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 5 . . 3  |-  F/ x  z  e.  A
43nfri 1530 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2297 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1362   F/wnf 1471    e. wcel 2160   F/_wnfc 2319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-cleq 2182  df-clel 2185  df-nfc 2321
This theorem is referenced by:  nfcri  2326
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