Theorem wl-sb8t 31950

Description: Substitution of variable in universal quantifier. Closed form of sb8 2273. (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 1999	. 2 |- F/ x A. x F/ y ph
2		nfnf1 2001	. . 3 |- F/ y F/ y ph
3	2	nfal 2049	. 2 |- F/ y A. x F/ y ph
4		sp 1957	. 2 |- ( A. x F/ y ph -> F/ y ph )
5		wl-nfs1t 31941	. . 3 |- ( F/ y ph -> F/ x [ y / x ] ph )
6	5	sps 1963	. 2 |- ( A. x F/ y ph -> F/ x [ y / x ] ph )
7		sbequ12 2098	. . 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 2126	1 |- ( A. x F/ y ph -> ( A. x ph <-> A. y [ y / x ] ph ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450   F/wnf 1675   [wsb 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806

This theorem is referenced by:  wl-sb8et  31951  wl-sbhbt  31952