Theorem sbmo 1959

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 1421	. . . . . 6 |- F/ x z = w
2	1	sblim 1831	. . . . 5 |- ( [ y / x ] ( ( ph /\ [ w / z ] ph ) -> z = w ) <-> ( [ y / x ] ( ph /\ [ w / z ] ph ) -> z = w ) )
3		sban 1829	. . . . . 6 |- ( [ y / x ] ( ph /\ [ w / z ] ph ) <-> ( [ y / x ] ph /\ [ y / x ] [ w / z ] ph ) )
4	3	imbi1i 227	. . . . 5 |- ( ( [ y / x ] ( ph /\ [ w / z ] ph ) -> z = w ) <-> ( ( [ y / x ] ph /\ [ y / x ] [ w / z ] ph ) -> z = w ) )
5		sbcom2 1863	. . . . . . 7 |- ( [ y / x ] [ w / z ] ph <-> [ w / z ] [ y / x ] ph )
6	5	anbi2i 430	. . . . . 6 |- ( ( [ y / x ] ph /\ [ y / x ] [ w / z ] ph ) <-> ( [ y / x ] ph /\ [ w / z ] [ y / x ] ph ) )
7	6	imbi1i 227	. . . . 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 195	. . . 4 |- ( [ y / x ] ( ( ph /\ [ w / z ] ph ) -> z = w ) <-> ( ( [ y / x ] ph /\ [ w / z ] [ y / x ] ph ) -> z = w ) )
9	8	sbalv 1881	. . 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 1881	. 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 1421	. . . 4 |- F/ w ph
12	11	mo3 1954	. . 3 |- ( E* z ph <-> A. z A. w ( ( ph /\ [ w / z ] ph ) -> z = w ) )
13	12	sbbii 1648	. 2 |- ( [ y / x ] E* z ph <-> [ y / x ] A. z A. w ( ( ph /\ [ w / z ] ph ) -> z = w ) )
14		nfv 1421	. . 3 |- F/ w [ y / x ] ph
15	14	mo3 1954	. 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 201	1 |- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   [wsb 1645   E*wmo 1901

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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by: (None)