Theorem sbequi 1832

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 1830	. . . 4 |- ( A. z z = x \/ F/ z [ x / z ] ph )
2		nfr 1511	. . . . . 6 |- ( F/ z [ x / z ] ph -> ( [ x / z ] ph -> A. z [ x / z ] ph ) )
3		equvini 1751	. . . . . . 7 |- ( x = y -> E. z ( x = z /\ z = y ) )
4		stdpc7 1763	. . . . . . . . 9 |- ( x = z -> ( [ x / z ] ph -> ph ) )
5		sbequ1 1761	. . . . . . . . 9 |- ( z = y -> ( ph -> [ y / z ] ph ) )
6	4, 5	sylan9 407	. . . . . . . 8 |- ( ( x = z /\ z = y ) -> ( [ x / z ] ph -> [ y / z ] ph ) )
7	6	eximi 1593	. . . . . . 7 |- ( E. z ( x = z /\ z = y ) -> E. z ( [ x / z ] ph -> [ y / z ] ph ) )
8		19.35-1 1617	. . . . . . 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 756	. . . 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 1830	. . . . 5 |- ( A. z z = y \/ F/ z [ y / z ] ph )
14		19.9t 1635	. . . . . . 7 |- ( F/ z [ y / z ] ph -> ( E. z [ y / z ] ph <-> [ y / z ] ph ) )
15	14	biimpd 143	. . . . . 6 |- ( F/ z [ y / z ] ph -> ( E. z [ y / z ] ph -> [ y / z ] ph ) )
16	15	orim2i 756	. . . . 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 756	. . . 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 1831	. 2 |- ( A. z z = x \/ ( A. z z = y \/ ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) ) )
22		sbequ2 1762	. . . . . . 7 |- ( z = x -> ( [ x / z ] ph -> ph ) )
23	22	sps 1530	. . . . . 6 |- ( A. z z = x -> ( [ x / z ] ph -> ph ) )
24	23	adantr 274	. . . . 5 |- ( ( A. z z = x /\ x = y ) -> ( [ x / z ] ph -> ph ) )
25		sbequ1 1761	. . . . . 6 |- ( x = y -> ( ph -> [ y / x ] ph ) )
26		drsb1 1792	. . . . . . . 8 |- ( A. x x = z -> ( [ y / x ] ph <-> [ y / z ] ph ) )
27	26	biimpd 143	. . . . . . 7 |- ( A. x x = z -> ( [ y / x ] ph -> [ y / z ] ph ) )
28	27	alequcoms 1509	. . . . . 6 |- ( A. z z = x -> ( [ y / x ] ph -> [ y / z ] ph ) )
29	25, 28	sylan9r 408	. . . . 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 114	. . 3 |- ( A. z z = x -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
32		drsb1 1792	. . . . . . . . 9 |- ( A. z z = y -> ( [ x / z ] ph <-> [ x / y ] ph ) )
33	32	biimpd 143	. . . . . . . 8 |- ( A. z z = y -> ( [ x / z ] ph -> [ x / y ] ph ) )
34		stdpc7 1763	. . . . . . . 8 |- ( x = y -> ( [ x / y ] ph -> ph ) )
35	33, 34	sylan9 407	. . . . . . 7 |- ( ( A. z z = y /\ x = y ) -> ( [ x / z ] ph -> ph ) )
36	5	sps 1530	. . . . . . . 8 |- ( A. z z = y -> ( ph -> [ y / z ] ph ) )
37	36	adantr 274	. . . . . . 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 114	. . . . 5 |- ( A. z z = y -> ( x = y -> ( [ x / z ] ph -> [ y / z ] ph ) ) )
40	39	orim1i 755	. . . 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		pm1.2 751	. . . 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 711	. 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 103   \/ wo 703   A.wal 1346   F/wnf 1453   E.wex 1485   [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by:  sbequ  1833