Theorem sbequi 1885

Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)

Assertion

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

Proof of Theorem sbequi

Step	Hyp	Ref	Expression
1		nfsb2or 1883	. . . 4 |- ( A. z z = x \/ F/ z [ x / z ] ph )
2		nfr 1564	. . . . . 6 |- ( F/ z [ x / z ] ph -> ( [ x / z ] ph -> A. z [ x / z ] ph ) )
3		equvini 1804	. . . . . . 7 |- ( x = y -> E. z ( x = z /\ z = y ) )
4		stdpc7 1816	. . . . . . . . 9 |- ( x = z -> ( [ x / z ] ph -> ph ) )
5		sbequ1 1814	. . . . . . . . 9 |- ( z = y -> ( ph -> [ y / z ] ph ) )
6	4, 5	sylan9 409	. . . . . . . 8 |- ( ( x = z /\ z = y ) -> ( [ x / z ] ph -> [ y / z ] ph ) )
7	6	eximi 1646	. . . . . . 7 |- ( E. z ( x = z /\ z = y ) -> E. z ( [ x / z ] ph -> [ y / z ] ph ) )
8		19.35-1 1670	. . . . . . 7 |- ( E. z ( [ x / z ] ph -> [ y / z ] ph ) -> ( A. z [ x / z ] ph -> E. z [ y / z ] ph ) )
9	3, 7, 8	3syl 17	. . . . . 6 |- ( x = y -> ( A. z [ x / z ] ph -> E. z [ y / z ] ph ) )
10	2, 9	syl9 72	. . . . 5 |- ( F/ z [ x / z ] ph -> ( x = y -> ( [ x / z ] ph -> E. z [ y / z ] ph ) ) )
11	10	orim2i 766	. . . 4 |- ( ( A. z z = x \/ F/ z [ x / z ] ph ) -> ( A. z z = x \/ ( x = y -> ( [ x / z ] ph -> E. z [ y / z ] ph ) ) ) )
12	1, 11	ax-mp 5	. . 3 |- ( A. z z = x \/ ( x = y -> ( [ x / z ] ph -> E. z [ y / z ] ph ) ) )
13		nfsb2or 1883	. . . . 5 |- ( A. z z = y \/ F/ z [ y / z ] ph )
14		19.9t 1688	. . . . . . 7 |- ( F/ z [ y / z ] ph -> ( E. z [ y / z ] ph <-> [ y / z ] ph ) )
15	14	biimpd 144	. . . . . 6 |- ( F/ z [ y / z ] ph -> ( E. z [ y / z ] ph -> [ y / z ] ph ) )
16	15	orim2i 766	. . . . 5 |- ( ( A. z z = y \/ F/ z [ y / z ] ph ) -> ( A. z z = y \/ ( E. z [ y / z ] ph -> [ y / z ] ph ) ) )
17	13, 16	ax-mp 5	. . . 4 |- ( A. z z = y \/ ( E. z [ y / z ] ph -> [ y / z ] ph ) )
18		ax-1 6	. . . . 5 |- ( ( E. z [ y / z ] ph -> [ y / z ] ph ) -> ( x = y -> ( E. z [ y / z ] ph -> [ y / z ] ph ) ) )
19	18	orim2i 766	. . . 4 |- ( ( A. z z = y \/ ( E. z [ y / z ] ph -> [ y / z ] ph ) ) -> ( A. z z = y \/ ( x = y -> ( E. z [ y / z ] ph -> [ y / z ] ph ) ) ) )
20	17, 19	ax-mp 5	. . 3 |- ( A. z z = y \/ ( x = y -> ( E. z [ y / z ] ph -> [ y / z ] ph ) ) )
21	12, 20	sbequilem 1884	. 2 |- ( A. z z = x \/ ( A. z z = y \/ ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) )
22		sbequ2 1815	. . . . . . 7 |- ( z = x -> ( [ x / z ] ph -> ph ) )
23	22	sps 1583	. . . . . 6 |- ( A. z z = x -> ( [ x / z ] ph -> ph ) )
24	23	adantr 276	. . . . 5 |- ( ( A. z z = x /\ x = y ) -> ( [ x / z ] ph -> ph ) )
25		sbequ1 1814	. . . . . 6 |- ( x = y -> ( ph -> [ y / x ] ph ) )
26		drsb1 1845	. . . . . . . 8 |- ( A. x x = z -> ( [ y / x ] ph <-> [ y / z ] ph ) )
27	26	biimpd 144	. . . . . . 7 |- ( A. x x = z -> ( [ y / x ] ph -> [ y / z ] ph ) )
28	27	alequcoms 1562	. . . . . 6 |- ( A. z z = x -> ( [ y / x ] ph -> [ y / z ] ph ) )
29	25, 28	sylan9r 410	. . . . 5 |- ( ( A. z z = x /\ x = y ) -> ( ph -> [ y / z ] ph ) )
30	24, 29	syld 45	. . . 4 |- ( ( A. z z = x /\ x = y ) -> ( [ x / z ] ph -> [ y / z ] ph ) )
31	30	ex 115	. . 3 |- ( A. z z = x -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
32		drsb1 1845	. . . . . . . . 9 |- ( A. z z = y -> ( [ x / z ] ph <-> [ x / y ] ph ) )
33	32	biimpd 144	. . . . . . . 8 |- ( A. z z = y -> ( [ x / z ] ph -> [ x / y ] ph ) )
34		stdpc7 1816	. . . . . . . 8 |- ( x = y -> ( [ x / y ] ph -> ph ) )
35	33, 34	sylan9 409	. . . . . . 7 |- ( ( A. z z = y /\ x = y ) -> ( [ x / z ] ph -> ph ) )
36	5	sps 1583	. . . . . . . 8 |- ( A. z z = y -> ( ph -> [ y / z ] ph ) )
37	36	adantr 276	. . . . . . 7 |- ( ( A. z z = y /\ x = y ) -> ( ph -> [ y / z ] ph ) )
38	35, 37	syld 45	. . . . . 6 |- ( ( A. z z = y /\ x = y ) -> ( [ x / z ] ph -> [ y / z ] ph ) )
39	38	ex 115	. . . . 5 |- ( A. z z = y -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
40	39	orim1i 765	. . . 4 |- ( ( A. z z = y \/ ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) -> ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) \/ ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) )
41		pm1.2 761	. . . 4 |- ( ( ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) \/ ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
42	40, 41	syl 14	. . 3 |- ( ( A. z z = y \/ ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
43	31, 42	jaoi 721	. 2 |- ( ( A. z z = x \/ ( A. z z = y \/ ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) ) -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
44	21, 43	ax-mp 5	1 |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   A.wal 1393   F/wnf 1506   E.wex 1538   [wsb 1808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809

This theorem is referenced by:  sbequ  1886