Theorem 19.9alt 34252

Description: Version of 19.9t 2071 for universal quantifier. (Contributed by NM, 9-Nov-2020.)

Assertion

Ref	Expression
19.9alt	|- ( F/ x ph -> ( A. x ph <-> ph ) )

Proof of Theorem 19.9alt

Step	Hyp	Ref	Expression
1		nfnt 1782	. . . 4 |- ( F/ x ph -> F/ x -. ph )
2		19.9t 2071	. . . 4 |- ( F/ x -. ph -> ( E. x -. ph <-> -. ph ) )
3	1, 2	syl 17	. . 3 |- ( F/ x ph -> ( E. x -. ph <-> -. ph ) )
4	3	con2bid 344	. 2 |- ( F/ x ph -> ( ph <-> -. E. x -. ph ) )
5		alex 1753	. 2 |- ( A. x ph <-> -. E. x -. ph )
6	4, 5	syl6rbbr 279	1 |- ( F/ x ph -> ( A. x ph <-> ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   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-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)