Theorem sbidm 1851

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 1763	. . . . 5 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
2	1	simplbi 274	. . . 4 |- ( [ y / x ] ph -> ( x = y -> ph ) )
3	2	sbimi 1764	. . 3 |- ( [ y / x ] [ y / x ] ph -> [ y / x ] ( x = y -> ph ) )
4		sbequ8 1847	. . 3 |- ( [ y / x ] ph <-> [ y / x ] ( x = y -> ph ) )
5	3, 4	sylibr 134	. 2 |- ( [ y / x ] [ y / x ] ph -> [ y / x ] ph )
6		ax-1 6	. . 3 |- ( [ y / x ] ph -> ( x = y -> [ y / x ] ph ) )
7		sb1 1766	. . . 4 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
8		pm4.24 395	. . . . . . . 8 |- ( E. x ( x = y /\ ph ) <-> ( E. x ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) )
9		ax-ie1 1493	. . . . . . . . 9 |- ( E. x ( x = y /\ ph ) -> A. x E. x ( x = y /\ ph ) )
10	9	19.41h 1685	. . . . . . . 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 187	. . . . . . 7 |- ( E. x ( x = y /\ ph ) <-> E. x ( ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) )
12		ax-1 6	. . . . . . . . . 10 |- ( ph -> ( x = y -> ph ) )
13	12	anim2i 342	. . . . . . . . 9 |- ( ( x = y /\ ph ) -> ( x = y /\ ( x = y -> ph ) ) )
14	13	anim1i 340	. . . . . . . 8 |- ( ( ( x = y /\ ph ) /\ E. x ( x = y /\ ph ) ) -> ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) )
15	14	eximi 1600	. . . . . . 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 121	. . . . . 6 |- ( E. x ( x = y /\ ph ) -> E. x ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) )
17		anass 401	. . . . . . 7 |- ( ( ( x = y /\ ( x = y -> ph ) ) /\ E. x ( x = y /\ ph ) ) <-> ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
18	17	exbii 1605	. . . . . 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 122	. . . . 5 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
20	1	anbi2i 457	. . . . . 6 |- ( ( x = y /\ [ y / x ] ph ) <-> ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
21	20	exbii 1605	. . . . 5 |- ( E. x ( x = y /\ [ y / x ] ph ) <-> E. x ( x = y /\ ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) ) )
22	19, 21	sylibr 134	. . . 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 1763	. . 3 |- ( [ y / x ] [ y / x ] ph <-> ( ( x = y -> [ y / x ] ph ) /\ E. x ( x = y /\ [ y / x ] ph ) ) )
25	6, 23, 24	sylanbrc 417	. 2 |- ( [ y / x ] ph -> [ y / x ] [ y / x ] ph )
26	5, 25	impbii 126	1 |- ( [ y / x ] [ y / x ] ph <-> [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1492   [wsb 1762

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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-sb 1763

This theorem is referenced by: (None)