Theorem sbequi 2059

Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
sbequi	|- (x = y -> ([x / z]ph -> [y / z]ph))

Proof of Theorem sbequi

Step	Hyp	Ref	Expression
1		hbsb2 2057	. . . . . 6 |- (-. A.z z = x -> ([x / z]ph -> A.z[x / z]ph))
2		equvini 1987	. . . . . . . 8 |- (x = y -> E.z(x = z /\ z = y))
3		stdpc7 1917	. . . . . . . . . 10 |- (x = z -> ([x / z]ph -> ph))
4		sbequ1 1918	. . . . . . . . . 10 |- (z = y -> (ph -> [y / z]ph))
5	3, 4	sylan9 638	. . . . . . . . 9 |- ((x = z /\ z = y) -> ([x / z]ph -> [y / z]ph))
6	5	eximi 1576	. . . . . . . 8 |- (E.z(x = z /\ z = y) -> E.z([x / z]ph -> [y / z]ph))
7	2, 6	syl 15	. . . . . . 7 |- (x = y -> E.z([x / z]ph -> [y / z]ph))
8		19.35 1600	. . . . . . 7 |- (E.z([x / z]ph -> [y / z]ph) <-> (A.z[x / z]ph -> E.z[y / z]ph))
9	7, 8	sylib 188	. . . . . 6 |- (x = y -> (A.z[x / z]ph -> E.z[y / z]ph))
10	1, 9	sylan9 638	. . . . 5 |- ((-. A.z z = x /\ x = y) -> ([x / z]ph -> E.z[y / z]ph))
11		nfsb2 2058	. . . . . 6 |- (-. A.z z = y -> F/z[y / z]ph)
12	11	19.9d 1782	. . . . 5 |- (-. A.z z = y -> (E.z[y / z]ph -> [y / z]ph))
13	10, 12	syl9 66	. . . 4 |- ((-. A.z z = x /\ x = y) -> (-. A.z z = y -> ([x / z]ph -> [y / z]ph)))
14	13	ex 423	. . 3 |- (-. A.z z = x -> (x = y -> (-. A.z z = y -> ([x / z]ph -> [y / z]ph))))
15	14	com23 72	. 2 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> ([x / z]ph -> [y / z]ph))))
16		sbequ2 1650	. . . . . 6 |- (z = x -> ([x / z]ph -> ph))
17	16	sps 1754	. . . . 5 |- (A.z z = x -> ([x / z]ph -> ph))
18	17	adantr 451	. . . 4 |- ((A.z z = x /\ x = y) -> ([x / z]ph -> ph))
19		sbequ1 1918	. . . . 5 |- (x = y -> (ph -> [y / x]ph))
20		drsb1 2022	. . . . . 6 |- (A.z z = x -> ([y / z]ph <-> [y / x]ph))
21	20	biimprd 214	. . . . 5 |- (A.z z = x -> ([y / x]ph -> [y / z]ph))
22	19, 21	sylan9r 639	. . . 4 |- ((A.z z = x /\ x = y) -> (ph -> [y / z]ph))
23	18, 22	syld 40	. . 3 |- ((A.z z = x /\ x = y) -> ([x / z]ph -> [y / z]ph))
24	23	ex 423	. 2 |- (A.z z = x -> (x = y -> ([x / z]ph -> [y / z]ph)))
25		drsb1 2022	. . . . . 6 |- (A.z z = y -> ([x / z]ph <-> [x / y]ph))
26	25	biimpd 198	. . . . 5 |- (A.z z = y -> ([x / z]ph -> [x / y]ph))
27		stdpc7 1917	. . . . 5 |- (x = y -> ([x / y]ph -> ph))
28	26, 27	sylan9 638	. . . 4 |- ((A.z z = y /\ x = y) -> ([x / z]ph -> ph))
29	4	sps 1754	. . . . 5 |- (A.z z = y -> (ph -> [y / z]ph))
30	29	adantr 451	. . . 4 |- ((A.z z = y /\ x = y) -> (ph -> [y / z]ph))
31	28, 30	syld 40	. . 3 |- ((A.z z = y /\ x = y) -> ([x / z]ph -> [y / z]ph))
32	31	ex 423	. 2 |- (A.z z = y -> (x = y -> ([x / z]ph -> [y / z]ph)))
33	15, 24, 32	pm2.61ii 157	1 |- (x = y -> ([x / z]ph -> [y / z]ph))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541  [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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbequ  2060