Theorem sbcom2 2114

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)

Assertion

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

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

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

Proof of Theorem sbcom2

Step	Hyp	Ref	Expression
1		alcom 1737	. . . . . 6 |- (A.zA.x(x = y -> (z = w -> ph)) <-> A.xA.z(x = y -> (z = w -> ph)))
2		bi2.04 350	. . . . . . . . 9 |- ((x = y -> (z = w -> ph)) <-> (z = w -> (x = y -> ph)))
3	2	albii 1566	. . . . . . . 8 |- (A.x(x = y -> (z = w -> ph)) <-> A.x(z = w -> (x = y -> ph)))
4		19.21v 1890	. . . . . . . 8 |- (A.x(z = w -> (x = y -> ph)) <-> (z = w -> A.x(x = y -> ph)))
5	3, 4	bitri 240	. . . . . . 7 |- (A.x(x = y -> (z = w -> ph)) <-> (z = w -> A.x(x = y -> ph)))
6	5	albii 1566	. . . . . 6 |- (A.zA.x(x = y -> (z = w -> ph)) <-> A.z(z = w -> A.x(x = y -> ph)))
7		19.21v 1890	. . . . . . 7 |- (A.z(x = y -> (z = w -> ph)) <-> (x = y -> A.z(z = w -> ph)))
8	7	albii 1566	. . . . . 6 |- (A.xA.z(x = y -> (z = w -> ph)) <-> A.x(x = y -> A.z(z = w -> ph)))
9	1, 6, 8	3bitr3i 266	. . . . 5 |- (A.z(z = w -> A.x(x = y -> ph)) <-> A.x(x = y -> A.z(z = w -> ph)))
10	9	a1i 10	. . . 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 2054	. . . . 5 |- (-. A.z z = w -> ([w / z][y / x]ph <-> A.z(z = w -> [y / x]ph)))
12		sb4b 2054	. . . . . . 7 |- (-. A.x x = y -> ([y / x]ph <-> A.x(x = y -> ph)))
13	12	imbi2d 307	. . . . . 6 |- (-. A.x x = y -> ((z = w -> [y / x]ph) <-> (z = w -> A.x(x = y -> ph))))
14	13	albidv 1625	. . . . 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 681	. . . 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 2054	. . . . 5 |- (-. A.x x = y -> ([y / x][w / z]ph <-> A.x(x = y -> [w / z]ph)))
17		sb4b 2054	. . . . . . 7 |- (-. A.z z = w -> ([w / z]ph <-> A.z(z = w -> ph)))
18	17	imbi2d 307	. . . . . 6 |- (-. A.z z = w -> ((x = y -> [w / z]ph) <-> (x = y -> A.z(z = w -> ph))))
19	18	albidv 1625	. . . . 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 680	. . . 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 276	. . 3 |- ((-. A.x x = y /\ -. A.z z = w) -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
22	21	ex 423	. 2 |- (-. A.x x = y -> (-. A.z z = w -> ([w / z][y / x]ph <-> [y / x][w / z]ph)))
23		nfae 1954	. . . 4 |- F/zA.x x = y
24		sbequ12 1919	. . . . 5 |- (x = y -> (ph <-> [y / x]ph))
25	24	sps 1754	. . . 4 |- (A.x x = y -> (ph <-> [y / x]ph))
26	23, 25	sbbid 2078	. . 3 |- (A.x x = y -> ([w / z]ph <-> [w / z][y / x]ph))
27		sbequ12 1919	. . . 4 |- (x = y -> ([w / z]ph <-> [y / x][w / z]ph))
28	27	sps 1754	. . 3 |- (A.x x = y -> ([w / z]ph <-> [y / x][w / z]ph))
29	26, 28	bitr3d 246	. 2 |- (A.x x = y -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
30		sbequ12 1919	. . . 4 |- (z = w -> ([y / x]ph <-> [w / z][y / x]ph))
31	30	sps 1754	. . 3 |- (A.z z = w -> ([y / x]ph <-> [w / z][y / x]ph))
32		nfae 1954	. . . 4 |- F/xA.z z = w
33		sbequ12 1919	. . . . 5 |- (z = w -> (ph <-> [w / z]ph))
34	33	sps 1754	. . . 4 |- (A.z z = w -> (ph <-> [w / z]ph))
35	32, 34	sbbid 2078	. . 3 |- (A.z z = w -> ([y / x]ph <-> [y / x][w / z]ph))
36	31, 35	bitr3d 246	. 2 |- (A.z z = w -> ([w / z][y / x]ph <-> [y / x][w / z]ph))
37	22, 29, 36	pm2.61ii 157	1 |- ([w / z][y / x]ph <-> [y / x][w / z]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:  2sb5rf  2117  2sb6rf  2118  2eu6  2289