Theorem wl-sb8t 33333

Description: Substitution of variable in universal quantifier. Closed form of sb8 2424. (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 2028	. 2 |- F/ x A. x F/ y ph
2		nfnf1 2031	. . 3 |- F/ y F/ y ph
3	2	nfal 2153	. 2 |- F/ y A. x F/ y ph
4		sp 2053	. 2 |- ( A. x F/ y ph -> F/ y ph )
5		wl-nfs1t 33324	. . 3 |- ( F/ y ph -> F/ x [ y / x ] ph )
6	5	sps 2055	. 2 |- ( A. x F/ y ph -> F/ x [ y / x ] ph )
7		sbequ12 2111	. . 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 2270	1 |- ( A. x F/ y ph -> ( A. x ph <-> A. y [ 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-sb8et  33334  wl-sbhbt  33335