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Theorem nfcrii 2571
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 2570 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 5 . . 3  |-  F/ x  z  e.  A
43nfri 1780 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2546 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550   F/wnf 1554    e. wcel 1727   F/_wnfc 2565
This theorem is referenced by:  nfcri  2572  abeq2f  23991  bnj1230  29272  bnj1000  29410  bnj1204  29479  bnj1307  29490  bnj1311  29491  bnj1398  29501  bnj1466  29520  bnj1467  29521  bnj1523  29538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-cleq 2435  df-clel 2438  df-nfc 2567
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