Theorem wl-sbhbt 33335

Description: Closed form of sbhb 2438. Characterizing the expression ph -> A. x ph using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)

Assertion

Ref	Expression
wl-sbhbt	|- ( A. x F/ y ph -> ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) )

Proof of Theorem wl-sbhbt

Step	Hyp	Ref	Expression
1		wl-sb8t 33333	. . 3 |- ( A. x F/ y ph -> ( A. x ph <-> A. y [ y / x ] ph ) )
2	1	imbi2d 330	. 2 |- ( A. x F/ y ph -> ( ( ph -> A. x ph ) <-> ( ph -> A. y [ y / x ] ph ) ) )
3		19.21t 2073	. . 3 |- ( F/ y ph -> ( A. y ( ph -> [ y / x ] ph ) <-> ( ph -> A. y [ y / x ] ph ) ) )
4	3	sps 2055	. 2 |- ( A. x F/ y ph -> ( A. y ( ph -> [ y / x ] ph ) <-> ( ph -> A. y [ y / x ] ph ) ) )
5	2, 4	bitr4d 271	1 |- ( A. x F/ y ph -> ( ( ph -> A. x ph ) <-> A. y ( ph -> [ y / x ] ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708   [wsb 1880

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-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  wl-sbnf1  33336