Theorem sbequi 1720

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 1718	. . . 4 |- ( A. z z = x \/ F/ z [ x / z ] ph )
2		nfr 1411	. . . . . 6 |- ( F/ z [ x / z ] ph -> ( [ x / z ] ph -> A. z [ x / z ] ph ) )
3		equvini 1641	. . . . . . 7 |- ( x = y -> E. z ( x = z /\ z = y ) )
4		stdpc7 1653	. . . . . . . . 9 |- ( x = z -> ( [ x / z ] ph -> ph ) )
5		sbequ1 1651	. . . . . . . . 9 |- ( z = y -> ( ph -> [ y / z ] ph ) )
6	4, 5	sylan9 389	. . . . . . . 8 |- ( ( x = z /\ z = y ) -> ( [ x / z ] ph -> [ y / z ] ph ) )
7	6	eximi 1491	. . . . . . 7 |- ( E. z ( x = z /\ z = y ) -> E. z ( [ x / z ] ph -> [ y / z ] ph ) )
8		19.35-1 1515	. . . . . . 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 66	. . . . 5 |- ( F/ z [ x / z ] ph -> ( x = y -> ( [ x / z ] ph -> E. z [ y / z ] ph ) ) )
11	10	orim2i 678	. . . 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 7	. . 3 |- ( A. z z = x \/ ( x = y -> ( [ x / z ] ph -> E. z [ y / z ] ph ) ) )
13		nfsb2or 1718	. . . . 5 |- ( A. z z = y \/ F/ z [ y / z ] ph )
14		19.9t 1533	. . . . . . 7 |- ( F/ z [ y / z ] ph -> ( E. z [ y / z ] ph <-> [ y / z ] ph ) )
15	14	biimpd 132	. . . . . 6 |- ( F/ z [ y / z ] ph -> ( E. z [ y / z ] ph -> [ y / z ] ph ) )
16	15	orim2i 678	. . . . 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 7	. . . 4 |- ( A. z z = y \/ ( E. z [ y / z ] ph -> [ y / z ] ph ) )
18		ax-1 5	. . . . 5 |- ( ( E. z [ y / z ] ph -> [ y / z ] ph ) -> ( x = y -> ( E. z [ y / z ] ph -> [ y / z ] ph ) ) )
19	18	orim2i 678	. . . 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 7	. . 3 |- ( A. z z = y \/ ( x = y -> ( E. z [ y / z ] ph -> [ y / z ] ph ) ) )
21	12, 20	sbequilem 1719	. 2 |- ( A. z z = x \/ ( A. z z = y \/ ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) )
22		sbequ2 1652	. . . . . . 7 |- ( z = x -> ( [ x / z ] ph -> ph ) )
23	22	sps 1430	. . . . . 6 |- ( A. z z = x -> ( [ x / z ] ph -> ph ) )
24	23	adantr 261	. . . . 5 |- ( ( A. z z = x /\ x = y ) -> ( [ x / z ] ph -> ph ) )
25		sbequ1 1651	. . . . . 6 |- ( x = y -> ( ph -> [ y / x ] ph ) )
26		drsb1 1680	. . . . . . . 8 |- ( A. x x = z -> ( [ y / x ] ph <-> [ y / z ] ph ) )
27	26	biimpd 132	. . . . . . 7 |- ( A. x x = z -> ( [ y / x ] ph -> [ y / z ] ph ) )
28	27	alequcoms 1409	. . . . . 6 |- ( A. z z = x -> ( [ y / x ] ph -> [ y / z ] ph ) )
29	25, 28	sylan9r 390	. . . . 5 |- ( ( A. z z = x /\ x = y ) -> ( ph -> [ y / z ] ph ) )
30	24, 29	syld 40	. . . 4 |- ( ( A. z z = x /\ x = y ) -> ( [ x / z ] ph -> [ y / z ] ph ) )
31	30	ex 108	. . 3 |- ( A. z z = x -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
32		drsb1 1680	. . . . . . . . 9 |- ( A. z z = y -> ( [ x / z ] ph <-> [ x / y ] ph ) )
33	32	biimpd 132	. . . . . . . 8 |- ( A. z z = y -> ( [ x / z ] ph -> [ x / y ] ph ) )
34		stdpc7 1653	. . . . . . . 8 |- ( x = y -> ( [ x / y ] ph -> ph ) )
35	33, 34	sylan9 389	. . . . . . 7 |- ( ( A. z z = y /\ x = y ) -> ( [ x / z ] ph -> ph ) )
36	5	sps 1430	. . . . . . . 8 |- ( A. z z = y -> ( ph -> [ y / z ] ph ) )
37	36	adantr 261	. . . . . . 7 |- ( ( A. z z = y /\ x = y ) -> ( ph -> [ y / z ] ph ) )
38	35, 37	syld 40	. . . . . 6 |- ( ( A. z z = y /\ x = y ) -> ( [ x / z ] ph -> [ y / z ] ph ) )
39	38	ex 108	. . . . 5 |- ( A. z z = y -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
40	39	orim1i 677	. . . 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		pm1.2 673	. . . 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 636	. 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 7	1 |- ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   A.wal 1241   F/wnf 1349   E.wex 1381   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbequ  1721