Theorem wl-sb8t 31844

Description: Substitution of variable in universal quantifier. Closed form of sb8 2222. (Contributed by Wolf Lammen, 27-Jul-2019.)

Assertion

Ref	Expression
wl-sb8t	|- ( A. x F/ y ph -> ( A. x ph <-> A. y [ y / x ] ph ) )

Proof of Theorem wl-sb8t

Step	Hyp	Ref	Expression
1		nfa1 1956	. 2 |- F/ x A. x F/ y ph
2		nfnf1 1958	. . 3 |- F/ y F/ y ph
3	2	nfal 2007	. 2 |- F/ y A. x F/ y ph
4		sp 1914	. 2 |- ( A. x F/ y ph -> F/ y ph )
5		wl-nfs1t 31835	. . 3 |- ( F/ y ph -> F/ x [ y / x ] ph )
6	5	sps 1920	. 2 |- ( A. x F/ y ph -> F/ x [ y / x ] ph )
7		sbequ12 2051	. . 3 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8	7	a1i 11	. 2 |- ( A. x F/ y ph -> ( x = y -> ( ph <-> [ y / x ] ph ) ) )
9	1, 3, 4, 6, 8	cbv2 2078	1 |- ( A. x F/ y ph -> ( A. x ph <-> A. y [ y / x ] ph ) )

Syntax hints:   -> wi 4   <-> wb 187   A.wal 1435   F/wnf 1661   [wsb 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057

This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791

This theorem is referenced by:  wl-sb8et  31845  wl-sbhbt  31846