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Theorem nfcrii 2516
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 2515 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 8 . . 3  |-  F/ x  z  e.  A
43nfri 1770 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2491 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546   F/wnf 1550    e. wcel 1717   F/_wnfc 2510
This theorem is referenced by:  nfcri  2517  abeq2f  23804  bnj1230  28512  bnj1000  28650  bnj1204  28719  bnj1307  28730  bnj1311  28731  bnj1398  28741  bnj1466  28760  bnj1467  28761  bnj1523  28778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2380  df-clel 2383  df-nfc 2512
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