Theorem sbal1 2126

Description: A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor -. A.xx = z. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sbal1	|- (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))

Distinct variable group:   x,y

Allowed substitution hints:   ph(x,y,z)

Proof of Theorem sbal1

Step	Hyp	Ref	Expression
1		sbequ12 1919	. . . . 5 |- (y = z -> (A.xph <-> [z / y]A.xph))
2	1	sps 1754	. . . 4 |- (A.y y = z -> (A.xph <-> [z / y]A.xph))
3		sbequ12 1919	. . . . . 6 |- (y = z -> (ph <-> [z / y]ph))
4	3	sps 1754	. . . . 5 |- (A.y y = z -> (ph <-> [z / y]ph))
5	4	dral2 1966	. . . 4 |- (A.y y = z -> (A.xph <-> A.x[z / y]ph))
6	2, 5	bitr3d 246	. . 3 |- (A.y y = z -> ([z / y]A.xph <-> A.x[z / y]ph))
7	6	a1d 22	. 2 |- (A.y y = z -> (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph)))
8		nfa1 1788	. . . . . . . 8 |- F/xA.xph
9	8	nfsb4 2081	. . . . . . 7 |- (-. A.x x = z -> F/x[z / y]A.xph)
10	9	nfrd 1763	. . . . . 6 |- (-. A.x x = z -> ([z / y]A.xph -> A.x[z / y]A.xph))
11		sp 1747	. . . . . . . 8 |- (A.xph -> ph)
12	11	sbimi 1652	. . . . . . 7 |- ([z / y]A.xph -> [z / y]ph)
13	12	alimi 1559	. . . . . 6 |- (A.x[z / y]A.xph -> A.x[z / y]ph)
14	10, 13	syl6 29	. . . . 5 |- (-. A.x x = z -> ([z / y]A.xph -> A.x[z / y]ph))
15	14	adantl 452	. . . 4 |- ((-. A.y y = z /\ -. A.x x = z) -> ([z / y]A.xph -> A.x[z / y]ph))
16		sb4 2053	. . . . . . . 8 |- (-. A.y y = z -> ([z / y]ph -> A.y(y = z -> ph)))
17	16	al2imi 1561	. . . . . . 7 |- (A.x -. A.y y = z -> (A.x[z / y]ph -> A.xA.y(y = z -> ph)))
18	17	hbnaes 1957	. . . . . 6 |- (-. A.y y = z -> (A.x[z / y]ph -> A.xA.y(y = z -> ph)))
19		ax-7 1734	. . . . . 6 |- (A.xA.y(y = z -> ph) -> A.yA.x(y = z -> ph))
20	18, 19	syl6 29	. . . . 5 |- (-. A.y y = z -> (A.x[z / y]ph -> A.yA.x(y = z -> ph)))
21		dveeq2 1940	. . . . . . . . 9 |- (-. A.x x = z -> (y = z -> A.x y = z))
22		alim 1558	. . . . . . . . 9 |- (A.x(y = z -> ph) -> (A.x y = z -> A.xph))
23	21, 22	syl9 66	. . . . . . . 8 |- (-. A.x x = z -> (A.x(y = z -> ph) -> (y = z -> A.xph)))
24	23	al2imi 1561	. . . . . . 7 |- (A.y -. A.x x = z -> (A.yA.x(y = z -> ph) -> A.y(y = z -> A.xph)))
25		sb2 2023	. . . . . . 7 |- (A.y(y = z -> A.xph) -> [z / y]A.xph)
26	24, 25	syl6 29	. . . . . 6 |- (A.y -. A.x x = z -> (A.yA.x(y = z -> ph) -> [z / y]A.xph))
27	26	hbnaes 1957	. . . . 5 |- (-. A.x x = z -> (A.yA.x(y = z -> ph) -> [z / y]A.xph))
28	20, 27	sylan9 638	. . . 4 |- ((-. A.y y = z /\ -. A.x x = z) -> (A.x[z / y]ph -> [z / y]A.xph))
29	15, 28	impbid 183	. . 3 |- ((-. A.y y = z /\ -. A.x x = z) -> ([z / y]A.xph <-> A.x[z / y]ph))
30	29	ex 423	. 2 |- (-. A.y y = z -> (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph)))
31	7, 30	pm2.61i 156	1 |- (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  [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:  sbal  2127