MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfcrd Structured version   Unicode version

Theorem nfcrd 2584
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfeqd.1  |-  ( ph  -> 
F/_ x A )
Assertion
Ref Expression
nfcrd  |-  ( ph  ->  F/ x  y  e.  A )
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem nfcrd
StepHypRef Expression
1 nfeqd.1 . 2  |-  ( ph  -> 
F/_ x A )
2 nfcr 2563 . 2  |-  ( F/_ x A  ->  F/ x  y  e.  A )
31, 2syl 16 1  |-  ( ph  ->  F/ x  y  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F/wnf 1553    e. wcel 1725   F/_wnfc 2558
This theorem is referenced by:  nfeqd  2585  nfeld  2586  dvelimdc  2591  nfcsbd  3276  nfifd  3754  axextnd  8458  axrepndlem1  8459  axunndlem1  8462  axregnd  8471  axextdist  25419
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-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551  df-nfc 2560
  Copyright terms: Public domain W3C validator