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Theorem nfcrii 2533
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 2532 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 8 . . 3  |-  F/ x  z  e.  A
43nfri 1774 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2508 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 1721   F/_wnfc 2527
This theorem is referenced by:  nfcri  2534  abeq2f  23913  bnj1230  28880  bnj1000  29018  bnj1204  29087  bnj1307  29098  bnj1311  29099  bnj1398  29109  bnj1466  29128  bnj1467  29129  bnj1523  29146
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2397  df-clel 2400  df-nfc 2529
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