Theorem sbidm 1772

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 1686	. . . . 5 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2	1	simplbi 268	. . . 4 |- ( [ y / x ] ph -> ( x = y -> ph ) )
3	2	sbimi 1687	. . 3 |- ( [ y / x ] [ y / x ] ph -> [ y / x ] ( x = y -> ph ) )
4		sbequ8 1768	. . 3 |- ( [ y / x ] ph <-> [ y / x ] ( x = y -> ph ) )
5	3, 4	sylibr 132	. 2 |- ( [ y / x ] [ y / x ] ph -> [ y / x ] ph )
6		ax-1 5	. . 3 |- ( [ y / x ] ph -> ( x = y -> [ y / x ] ph ) )
7		sb1 1689	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
8		pm4.24 387	. . . . . . . 8 |- ( E. x ( x = y /\ ph ) <-> ( E. x ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) )
9		ax-ie1 1422	. . . . . . . . 9 |- ( E. x ( x = y /\ ph ) -> A. x E. x ( x = y /\ ph ) )
10	9	19.41h 1615	. . . . . . . 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 185	. . . . . . 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 334	. . . . . . . . 9 |- ( ( x = y /\ ph ) -> ( x = y /\ ( x = y -> ph ) ) )
14	13	anim1i 333	. . . . . . . 8 |- ( ( ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) -> ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) )
15	14	eximi 1531	. . . . . . 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 119	. . . . . 6 |- ( E. x ( x = y /\ ph ) -> E. x ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) )
17		anass 393	. . . . . . 7 |- ( ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) <-> ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
18	17	exbii 1536	. . . . . 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 120	. . . . 5 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
20	1	anbi2i 444	. . . . . 6 |- ( ( x = y /\ [ y / x ] ph ) <-> ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
21	20	exbii 1536	. . . . 5 |- ( E. x ( x = y /\ [ y / x ] ph ) <-> E. x ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
22	19, 21	sylibr 132	. . . 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 1686	. . 3 |- ( [ y / x ] [ y / x ] ph <-> ( ( x = y -> [ y / x ] ph ) /\ E. x ( x = y /\ [ y / x ] ph ) ) )
25	6, 23, 24	sylanbrc 408	. 2 |- ( [ y / x ] ph -> [ y / x ] [ y / x ] ph )
26	5, 25	impbii 124	1 |- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1421   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by: (None)