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Theorem nfcrii 2414
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
Dummy variable  z is distinct from all other variables.
Allowed substitution hints:    A( x, y)

Proof of Theorem nfcrii
StepHypRef Expression
1 nfcri.1 . . . 4  |-  F/_ x A
2 nfcr 2413 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 10 . . 3  |-  F/ x  z  e.  A
43nfri 1744 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2389 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1528   F/wnf 1532    e. wcel 1685   F/_wnfc 2408
This theorem is referenced by:  nfcri  2415  bnj1230  28103  bnj1000  28241  bnj1204  28310  bnj1307  28321  bnj1311  28322  bnj1398  28332  bnj1466  28351  bnj1467  28352  bnj1523  28369
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-cleq 2278  df-clel 2281  df-nfc 2410
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