Theorem wl-nfs1t 33324

Description: If y is not free in ph, x is not free in [ y / x ] ph. Closed form of nfs1 2365. (Contributed by Wolf Lammen, 27-Jul-2019.)

Assertion

Ref	Expression
wl-nfs1t	|- ( F/ y ph -> F/ x [ y / x ] ph )

Proof of Theorem wl-nfs1t

Step	Hyp	Ref	Expression
1		sbequ12r 2112	. . . . . 6 |- ( y = x -> ( [ y / x ] ph <-> ph ) )
2	1	equcoms 1947	. . . . 5 |- ( x = y -> ( [ y / x ] ph <-> ph ) )
3	2	sps 2055	. . . 4 |- ( A. x x = y -> ( [ y / x ] ph <-> ph ) )
4	3	drnf1 2329	. . 3 |- ( A. x x = y -> ( F/ x [ y / x ] ph <-> F/ y ph ) )
5	4	biimprd 238	. 2 |- ( A. x x = y -> ( F/ y ph -> F/ x [ y / x ] ph ) )
6		nfsb2 2360	. . 3 |- ( -. A. x x = y -> F/ x [ y / x ] ph )
7	6	a1d 25	. 2 |- ( -. A. x x = y -> ( F/ y ph -> F/ x [ y / x ] ph ) )
8	5, 7	pm2.61i 176	1 |- ( F/ y ph -> F/ x [ y / x ] ph )

Syntax hints:   -. wn 3   -> 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:  wl-sb8t  33333