Theorem sbcom2 1373

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).

Assertion

Ref	Expression
sbcom2	|- ([w / z][y / x]ph <-> [y / x][w / z]ph)

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

Proof of Theorem sbcom2

Step	Hyp	Ref	Expression
1		alcom 1068	. . . . . 6 |- (A.zA.x(x = y -> (z = w -> ph)) <-> A.xA.z(x = y -> (z = w -> ph)))
2		bi2.04 158	. . . . . . . . 9 |- ((x = y -> (z = w -> ph)) <-> (z = w -> (x = y -> ph)))
3	2	albii 1035	. . . . . . . 8 |- (A.x(x = y -> (z = w -> ph)) <-> A.x(z = w -> (x = y -> ph)))
4		19.21v 1323	. . . . . . . 8 |- (A.x(z = w -> (x = y -> ph)) <-> (z = w -> A.x(x = y -> ph)))
5	3, 4	bitri 171	. . . . . . 7 |- (A.x(x = y -> (z = w -> ph)) <-> (z = w -> A.x(x = y -> ph)))
6	5	albii 1035	. . . . . 6 |- (A.zA.x(x = y -> (z = w -> ph)) <-> A.z(z = w -> A.x(x = y -> ph)))
7		19.21v 1323	. . . . . . 7 |- (A.z(x = y -> (z = w -> ph)) <-> (x = y -> A.z(z = w -> ph)))
8	7	albii 1035	. . . . . 6 |- (A.xA.z(x = y -> (z = w -> ph)) <-> A.x(x = y -> A.z(z = w -> ph)))
9	1, 6, 8	3bitr3i 179	. . . . 5 |- (A.z(z = w -> A.x(x = y -> ph)) <-> A.x(x = y -> A.z(z = w -> ph)))
10	9	a1i 8	. . . 4 |- ((-. A.x x = y /\ -. A.z z = w) -> (A.z(z = w -> A.x(x = y -> ph)) <-> A.x(x = y -> A.z(z = w -> ph))))
11		sb4b 1261	. . . . 5 |- (-. A.z z = w -> ([w / z][y / x]ph <-> A.z(z = w -> [y / x]ph)))
12		sb4b 1261	. . . . . . 7 |- (-. A.x x = y -> ([y / x]ph <-> A.x(x = y -> ph)))
13	12	imbi2d 615	. . . . . 6 |- (-. A.x x = y -> ((z = w -> [y / x]ph) <-> (z = w -> A.x(x = y -> ph))))
14	13	albidv 1316	. . . . 5 |- (-. A.x x = y -> (A.z(z = w -> [y / x]ph) <-> A.z(z = w -> A.x(x = y -> ph))))
15	11, 14	sylan9bbr 544	. . . 4 |- ((-. A.x x = y /\ -. A.z z = w) -> ([w / z][y / x]ph <-> A.z(z = w -> A.x(x = y -> ph))))
16		sb4b 1261	. . . . 5 |- (-. A.x x = y -> ([y / x][w / z]ph <-> A.x(x = y -> [w / z]ph)))
17		sb4b 1261	. . . . . . 7 |- (-. A.z z = w -> ([w / z]ph <-> A.z(z = w -> ph)))
18	17	imbi2d 615	. . . . . 6 |- (-. A.z z = w -> ((x = y -> [w / z]ph) <-> (x = y -> A.z(z = w -> ph))))
19	18	albidv 1316	. . . . 5 |- (-. A.z z = w -> (A.x(x = y -> [w / z]ph) <-> A.x(x = y -> A.z(z = w -> ph))))
20	16, 19	sylan9bb 543	. . . 4 |- ((-. A.x x = y /\ -. A.z z = w) -> ([y / x][w / z]ph <-> A.x(x = y -> A.z(z = w -> ph))))
21	10, 15, 20	3bitr4d 553	. . 3 |- ((-. A.x x = y /\ -. A.z z = w) -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
22	21	ex 371	. 2 |- (-. A.x x = y -> (-. A.z z = w -> ([w / z][y / x]ph <-> [y / x][w / z]ph)))
23		hbae 1182	. . . 4 |- (A.x x = y -> A.zA.x x = y)
24		sbequ12 1218	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
25	24	a4s 1020	. . . 4 |- (A.x x = y -> (ph <-> [y / x]ph))
26	23, 25	sbbid 1283	. . 3 |- (A.x x = y -> ([w / z]ph <-> [w / z][y / x]ph))
27		sbequ12 1218	. . . 4 |- (x = y -> ([w / z]ph <-> [y / x][w / z]ph))
28	27	a4s 1020	. . 3 |- (A.x x = y -> ([w / z]ph <-> [y / x][w / z]ph))
29	26, 28	bitr3d 533	. 2 |- (A.x x = y -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
30		sbequ12 1218	. . . 4 |- (z = w -> ([y / x]ph <-> [w / z][y / x]ph))
31	30	a4s 1020	. . 3 |- (A.z z = w -> ([y / x]ph <-> [w / z][y / x]ph))
32		hbae 1182	. . . 4 |- (A.z z = w -> A.xA.z z = w)
33		sbequ12 1218	. . . . 5 |- (z = w -> (ph <-> [w / z]ph))
34	33	a4s 1020	. . . 4 |- (A.z z = w -> (ph <-> [w / z]ph))
35	32, 34	sbbid 1283	. . 3 |- (A.z z = w -> ([y / x]ph <-> [y / x][w / z]ph))
36	31, 35	bitr3d 533	. 2 |- (A.z z = w -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
37	22, 29, 36	pm2.61ii 128	1 |- ([w / z][y / x]ph <-> [y / x][w / z]ph)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990  [wsbc 1207

This theorem is referenced by:  2sb5rf 1377  2sb6rf 1378  2eu6 1494

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209

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