Theorem sbnf2 2108

Description: Two ways of expressing "x is (effectively) not free in ph." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)

Assertion

Ref	Expression
sbnf2	|- ( F/xph <-> A.yA.z([y / x]ph <-> [z / x]ph))

Distinct variable groups:   x,y,z   ph,y,z

Allowed substitution hint:   ph(x)

Proof of Theorem sbnf2

Step	Hyp	Ref	Expression
1		2albiim 1612	. 2 |- (A.yA.z([y / x]ph <-> [z / x]ph) <-> (A.yA.z([y / x]ph -> [z / x]ph) /\ A.yA.z([z / x]ph -> [y / x]ph)))
2		df-nf 1545	. . . . 5 |- ( F/xph <-> A.x(ph -> A.xph))
3		sbhb 2107	. . . . . 6 |- ((ph -> A.xph) <-> A.z(ph -> [z / x]ph))
4	3	albii 1566	. . . . 5 |- (A.x(ph -> A.xph) <-> A.xA.z(ph -> [z / x]ph))
5		alcom 1737	. . . . 5 |- (A.xA.z(ph -> [z / x]ph) <-> A.zA.x(ph -> [z / x]ph))
6	2, 4, 5	3bitri 262	. . . 4 |- ( F/xph <-> A.zA.x(ph -> [z / x]ph))
7		nfv 1619	. . . . . . 7 |- F/y(ph -> [z / x]ph)
8	7	sb8 2092	. . . . . 6 |- (A.x(ph -> [z / x]ph) <-> A.y[y / x](ph -> [z / x]ph))
9		nfs1v 2106	. . . . . . . 8 |- F/x[z / x]ph
10	9	sblim 2068	. . . . . . 7 |- ([y / x](ph -> [z / x]ph) <-> ([y / x]ph -> [z / x]ph))
11	10	albii 1566	. . . . . 6 |- (A.y[y / x](ph -> [z / x]ph) <-> A.y([y / x]ph -> [z / x]ph))
12	8, 11	bitri 240	. . . . 5 |- (A.x(ph -> [z / x]ph) <-> A.y([y / x]ph -> [z / x]ph))
13	12	albii 1566	. . . 4 |- (A.zA.x(ph -> [z / x]ph) <-> A.zA.y([y / x]ph -> [z / x]ph))
14		alcom 1737	. . . 4 |- (A.zA.y([y / x]ph -> [z / x]ph) <-> A.yA.z([y / x]ph -> [z / x]ph))
15	6, 13, 14	3bitri 262	. . 3 |- ( F/xph <-> A.yA.z([y / x]ph -> [z / x]ph))
16		sbhb 2107	. . . . . 6 |- ((ph -> A.xph) <-> A.y(ph -> [y / x]ph))
17	16	albii 1566	. . . . 5 |- (A.x(ph -> A.xph) <-> A.xA.y(ph -> [y / x]ph))
18		alcom 1737	. . . . 5 |- (A.xA.y(ph -> [y / x]ph) <-> A.yA.x(ph -> [y / x]ph))
19	2, 17, 18	3bitri 262	. . . 4 |- ( F/xph <-> A.yA.x(ph -> [y / x]ph))
20		nfv 1619	. . . . . . 7 |- F/z(ph -> [y / x]ph)
21	20	sb8 2092	. . . . . 6 |- (A.x(ph -> [y / x]ph) <-> A.z[z / x](ph -> [y / x]ph))
22		nfs1v 2106	. . . . . . . 8 |- F/x[y / x]ph
23	22	sblim 2068	. . . . . . 7 |- ([z / x](ph -> [y / x]ph) <-> ([z / x]ph -> [y / x]ph))
24	23	albii 1566	. . . . . 6 |- (A.z[z / x](ph -> [y / x]ph) <-> A.z([z / x]ph -> [y / x]ph))
25	21, 24	bitri 240	. . . . 5 |- (A.x(ph -> [y / x]ph) <-> A.z([z / x]ph -> [y / x]ph))
26	25	albii 1566	. . . 4 |- (A.yA.x(ph -> [y / x]ph) <-> A.yA.z([z / x]ph -> [y / x]ph))
27	19, 26	bitri 240	. . 3 |- ( F/xph <-> A.yA.z([z / x]ph -> [y / x]ph))
28	15, 27	anbi12i 678	. 2 |- (( F/xph /\ F/xph) <-> (A.yA.z([y / x]ph -> [z / x]ph) /\ A.yA.z([z / x]ph -> [y / x]ph)))
29		anidm 625	. 2 |- (( F/xph /\ F/xph) <-> F/xph)
30	1, 28, 29	3bitr2ri 265	1 |- ( F/xph <-> A.yA.z([y / x]ph <-> [z / x]ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbnfc2  3197