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Theorem nfcrii 2221
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 2220 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 7 . . 3  |-  F/ x  z  e.  A
43nfri 1457 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2195 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287   F/wnf 1394    e. wcel 1438   F/_wnfc 2215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217
This theorem is referenced by:  nfcri  2222
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