Theorem nf5r 2064

Description: Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.)

Assertion

Ref	Expression
nf5r	|- ( F/ x ph -> ( ph -> A. x ph ) )

Proof of Theorem nf5r

Step	Hyp	Ref	Expression
1		19.8a 2052	. 2 |- ( ph -> E. x ph )
2		df-nf 1710	. . 3 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
3	2	biimpi 206	. 2 |- ( F/ x ph -> ( E. x ph -> A. x ph ) )
4	1, 3	syl5 34	1 |- ( F/ x ph -> ( ph -> A. x ph ) )

Syntax hints:   -> wi 4   A.wal 1481   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  nf5ri  2065  nf5rd  2066  19.21tOLDOLD  2074  sbft  2379  bj-alrim  32683  bj-nexdt  32687  bj-cbv3tb  32711  bj-nfs1t2  32715  bj-sbftv  32763  bj-equsal1t  32809  stdpc5t  32814  bj-axc14  32839  wl-nfeqfb  33323