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Theorem nfcrii 2385
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
StepHypRef Expression
1 nfcri.1 . . . 4  |-  F/_ x A
2 nfcr 2384 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 10 . . 3  |-  F/ x  z  e.  A
43nfri 1703 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2360 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6   A.wal 1532   F/wnf 1539    e. wcel 1621   F/_wnfc 2379
This theorem is referenced by:  nfcri  2386  bnj1230  27868  bnj1000  28006  bnj1204  28075  bnj1307  28086  bnj1311  28087  bnj1398  28097  bnj1466  28116  bnj1467  28117  bnj1523  28134
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-cleq 2249  df-clel 2252  df-nfc 2381
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