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Theorem nfcrii 2559
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 2558 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 8 . . 3  |-  F/ x  z  e.  A
43nfri 1778 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2534 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   F/wnf 1553    e. wcel 1725   F/_wnfc 2553
This theorem is referenced by:  nfcri  2560  abeq2f  23943  bnj1230  28926  bnj1000  29064  bnj1204  29133  bnj1307  29144  bnj1311  29145  bnj1398  29155  bnj1466  29174  bnj1467  29175  bnj1523  29192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-cleq 2423  df-clel 2426  df-nfc 2555
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