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Theorem nfr 1777
Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.)
Assertion
Ref Expression
nfr  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)

Proof of Theorem nfr
StepHypRef Expression
1 df-nf 1554 . 2  |-  ( F/ x ph  <->  A. x
( ph  ->  A. x ph ) )
2 sp 1763 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( ph  ->  A. x ph ) )
31, 2sylbi 188 1  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   F/wnf 1553
This theorem is referenced by:  nfri  1778  nfrd  1779  19.21t  1813  19.23t  1818  nfimd  1827  nfaldOLD  1872  spimtOLD  1956  sbft  2103  sbftOLD  2104  nfaldwAUX7  29353  spimtNEW7  29408  sbftNEW7  29457  ax7wnftAUX7  29558  nfaldOLD7  29591
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-nf 1554
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