ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfcrii Unicode version

Theorem nfcrii 2213
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 2212 . . . 4  |-  ( F/_ x A  ->  F/ x  z  e.  A )
31, 2ax-mp 7 . . 3  |-  F/ x  z  e.  A
43nfri 1453 . 2  |-  ( z  e.  A  ->  A. x  z  e.  A )
54hblem 2187 1  |-  ( y  e.  A  ->  A. x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   F/wnf 1390    e. wcel 1434   F/_wnfc 2207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078  df-nfc 2209
This theorem is referenced by:  nfcri  2214
  Copyright terms: Public domain W3C validator