Theorem sbmo 2354

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		sbex 2302	. . 3 |- ( [ y / x ] E. w A. z ( ph -> z = w ) <-> E. w [ y / x ] A. z ( ph -> z = w ) )
2		nfv 1771	. . . . . 6 |- F/ x z = w
3	2	sblim 2236	. . . . 5 |- ( [ y / x ] ( ph -> z = w ) <-> ( [ y / x ] ph -> z = w ) )
4	3	sbalv 2303	. . . 4 |- ( [ y / x ] A. z ( ph -> z = w ) <-> A. z ( [ y / x ] ph -> z = w ) )
5	4	exbii 1728	. . 3 |- ( E. w [ y / x ] A. z ( ph -> z = w ) <-> E. w A. z ( [ y / x ] ph -> z = w ) )
6	1, 5	bitri 257	. 2 |- ( [ y / x ] E. w A. z ( ph -> z = w ) <-> E. w A. z ( [ y / x ] ph -> z = w ) )
7		mo2v 2316	. . 3 |- ( E* z ph <-> E. w A. z ( ph -> z = w ) )
8	7	sbbii 1814	. 2 |- ( [ y / x ] E* z ph <-> [ y / x ] E. w A. z ( ph -> z = w ) )
9		mo2v 2316	. 2 |- ( E* z [ y / x ] ph <-> E. w A. z ( [ y / x ] ph -> z = w ) )
10	6, 8, 9	3bitr4i 285	1 |- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1452   E.wex 1673   [wsb 1807   E*wmo 2310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314

This theorem is referenced by: (None)