Theorem bj-ssbft 32642

Description: See sbft 2379. This proof is from Tarski's FOL together with sp 2053 (and its dual). (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbft	|- ( F/ x ph -> ([ t/ x]b ph <-> ph ) )

Proof of Theorem bj-ssbft

Step	Hyp	Ref	Expression
1		bj-sbex 32626	. . 3 |- ([ t/ x]b ph -> E. x ph )
2		df-nf 1710	. . . 4 |- ( F/ x ph <-> ( E. x ph -> A. x ph ) )
3	2	biimpi 206	. . 3 |- ( F/ x ph -> ( E. x ph -> A. x ph ) )
4		sp 2053	. . 3 |- ( A. x ph -> ph )
5	1, 3, 4	syl56 36	. 2 |- ( F/ x ph -> ([ t/ x]b ph -> ph ) )
6		19.8a 2052	. . 3 |- ( ph -> E. x ph )
7		bj-alsb 32625	. . 3 |- ( A. x ph -> [ t/ x]b ph )
8	6, 3, 7	syl56 36	. 2 |- ( F/ x ph -> ( ph -> [ t/ x]b ph ) )
9	5, 8	impbid 202	1 |- ( F/ x ph -> ([ t/ x]b ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708  [wssb 32619

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  df-ssb 32620

This theorem is referenced by: (None)