Theorem sbidm 1731

Description: An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)

Assertion

Ref	Expression
sbidm	|- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph )

Proof of Theorem sbidm

Step	Hyp	Ref	Expression
1		df-sb 1646	. . . . 5 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2	1	simplbi 259	. . . 4 |- ( [ y / x ] ph -> ( x = y -> ph ) )
3	2	sbimi 1647	. . 3 |- ( [ y / x ] [ y / x ] ph -> [ y / x ] ( x = y -> ph ) )
4		sbequ8 1727	. . 3 |- ( [ y / x ] ph <-> [ y / x ] ( x = y -> ph ) )
5	3, 4	sylibr 137	. 2 |- ( [ y / x ] [ y / x ] ph -> [ y / x ] ph )
6		ax-1 5	. . 3 |- ( [ y / x ] ph -> ( x = y -> [ y / x ] ph ) )
7		sb1 1649	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
8		pm4.24 375	. . . . . . . 8 |- ( E. x ( x = y /\ ph ) <-> ( E. x ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) )
9		ax-ie1 1382	. . . . . . . . 9 |- ( E. x ( x = y /\ ph ) -> A. x E. x ( x = y /\ ph ) )
10	9	19.41h 1575	. . . . . . . 8 |- ( E. x ( ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) <-> ( E. x ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) )
11	8, 10	bitr4i 176	. . . . . . 7 |- ( E. x ( x = y /\ ph ) <-> E. x ( ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) )
12		ax-1 5	. . . . . . . . . 10 |- ( ph -> ( x = y -> ph ) )
13	12	anim2i 324	. . . . . . . . 9 |- ( ( x = y /\ ph ) -> ( x = y /\ ( x = y -> ph ) ) )
14	13	anim1i 323	. . . . . . . 8 |- ( ( ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) -> ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) )
15	14	eximi 1491	. . . . . . 7 |- ( E. x ( ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) -> E. x ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) )
16	11, 15	sylbi 114	. . . . . 6 |- ( E. x ( x = y /\ ph ) -> E. x ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) )
17		anass 381	. . . . . . 7 |- ( ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) <-> ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
18	17	exbii 1496	. . . . . 6 |- ( E. x ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) <-> E. x ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
19	16, 18	sylib 127	. . . . 5 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
20	1	anbi2i 430	. . . . . 6 |- ( ( x = y /\ [ y / x ] ph ) <-> ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
21	20	exbii 1496	. . . . 5 |- ( E. x ( x = y /\ [ y / x ] ph ) <-> E. x ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
22	19, 21	sylibr 137	. . . 4 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ [ y / x ] ph ) )
23	7, 22	syl 14	. . 3 |- ( [ y / x ] ph -> E. x ( x = y /\ [ y / x ] ph ) )
24		df-sb 1646	. . 3 |- ( [ y / x ] [ y / x ] ph <-> ( ( x = y -> [ y / x ] ph ) /\ E. x ( x = y /\ [ y / x ] ph ) ) )
25	6, 23, 24	sylanbrc 394	. 2 |- ( [ y / x ] ph -> [ y / x ] [ y / x ] ph )
26	5, 25	impbii 117	1 |- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

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

This theorem is referenced by: (None)