Theorem sbmo 2000

Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
sbmo	|- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem sbmo

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . . . 6 |- F/ x z = w
2	1	sblim 1872	. . . . 5 |- ( [ y / x ] ( ( ph /\ [ w / z ] ph ) -> z = w ) <-> ( [ y / x ] ( ph /\ [ w / z ] ph ) -> z = w ) )
3		sban 1870	. . . . . 6 |- ( [ y / x ] ( ph /\ [ w / z ] ph ) <-> ( [ y / x ] ph /\ [ y / x ] [ w / z ] ph ) )
4	3	imbi1i 236	. . . . 5 |- ( ( [ y / x ] ( ph /\ [ w / z ] ph ) -> z = w ) <-> ( ( [ y / x ] ph /\ [ y / x ] [ w / z ] ph ) -> z = w ) )
5		sbcom2 1904	. . . . . . 7 |- ( [ y / x ] [ w / z ] ph <-> [ w / z ] [ y / x ] ph )
6	5	anbi2i 444	. . . . . 6 |- ( ( [ y / x ] ph /\ [ y / x ] [ w / z ] ph ) <-> ( [ y / x ] ph /\ [ w / z ] [ y / x ] ph ) )
7	6	imbi1i 236	. . . . 5 |- ( ( ( [ y / x ] ph /\ [ y / x ] [ w / z ] ph ) -> z = w ) <-> ( ( [ y / x ] ph /\ [ w / z ] [ y / x ] ph ) -> z = w ) )
8	2, 4, 7	3bitri 204	. . . 4 |- ( [ y / x ] ( ( ph /\ [ w / z ] ph ) -> z = w ) <-> ( ( [ y / x ] ph /\ [ w / z ] [ y / x ] ph ) -> z = w ) )
9	8	sbalv 1922	. . 3 |- ( [ y / x ] A. w ( ( ph /\ [ w / z ] ph ) -> z = w ) <-> A. w ( ( [ y / x ] ph /\ [ w / z ] [ y / x ] ph ) -> z = w ) )
10	9	sbalv 1922	. 2 |- ( [ y / x ] A. z A. w ( ( ph /\ [ w / z ] ph ) -> z = w ) <-> A. z A. w ( ( [ y / x ] ph /\ [ w / z ] [ y / x ] ph ) -> z = w ) )
11		nfv 1461	. . . 4 |- F/ w ph
12	11	mo3 1995	. . 3 |- ( E* z ph <-> A. z A. w ( ( ph /\ [ w / z ] ph ) -> z = w ) )
13	12	sbbii 1688	. 2 |- ( [ y / x ] E* z ph <-> [ y / x ] A. z A. w ( ( ph /\ [ w / z ] ph ) -> z = w ) )
14		nfv 1461	. . 3 |- F/ w [ y / x ] ph
15	14	mo3 1995	. 2 |- ( E* z [ y / x ] ph <-> A. z A. w ( ( [ y / x ] ph /\ [ w / z ] [ y / x ] ph ) -> z = w ) )
16	10, 13, 15	3bitr4i 210	1 |- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   [wsb 1685   E*wmo 1942

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by: (None)