Theorem wl-sb6rft 33330

Description: A specialization of wl-equsal1t 33327. Closed form of sb6rf 2423. (Contributed by Wolf Lammen, 27-Jul-2019.)

Assertion

Ref	Expression
wl-sb6rft	|- ( F/ x ph -> ( ph <-> A. x ( x = y -> [ x / y ] ph ) ) )

Proof of Theorem wl-sb6rft

Step	Hyp	Ref	Expression
1		nfnf1 2031	. . 3 |- F/ x F/ x ph
2		id 22	. . 3 |- ( F/ x ph -> F/ x ph )
3		sbequ12r 2112	. . . 4 |- ( x = y -> ( [ x / y ] ph <-> ph ) )
4	3	a1i 11	. . 3 |- ( F/ x ph -> ( x = y -> ( [ x / y ] ph <-> ph ) ) )
5	1, 2, 4	wl-equsald 33325	. 2 |- ( F/ x ph -> ( A. x ( x = y -> [ x / y ] ph ) <-> ph ) )
6	5	bicomd 213	1 |- ( F/ x ph -> ( ph <-> A. x ( x = y -> [ x / y ] 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-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: (None)