Theorem sbmo 2515

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 2463	. . 3 |- ( [ y / x ] E. w A. z ( ph -> z = w ) <-> E. w [ y / x ] A. z ( ph -> z = w ) )
2		nfv 1843	. . . . . 6 |- F/ x z = w
3	2	sblim 2397	. . . . 5 |- ( [ y / x ] ( ph -> z = w ) <-> ( [ y / x ] ph -> z = w ) )
4	3	sbalv 2464	. . . 4 |- ( [ y / x ] A. z ( ph -> z = w ) <-> A. z ( [ y / x ] ph -> z = w ) )
5	4	exbii 1774	. . 3 |- ( E. w [ y / x ] A. z ( ph -> z = w ) <-> E. w A. z ( [ y / x ] ph -> z = w ) )
6	1, 5	bitri 264	. 2 |- ( [ y / x ] E. w A. z ( ph -> z = w ) <-> E. w A. z ( [ y / x ] ph -> z = w ) )
7		mo2v 2477	. . 3 |- ( E* z ph <-> E. w A. z ( ph -> z = w ) )
8	7	sbbii 1887	. 2 |- ( [ y / x ] E* z ph <-> [ y / x ] E. w A. z ( ph -> z = w ) )
9		mo2v 2477	. 2 |- ( E* z [ y / x ] ph <-> E. w A. z ( [ y / x ] ph -> z = w ) )
10	6, 8, 9	3bitr4i 292	1 |- ( [ y / x ] E* z ph <-> E* z [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   [wsb 1880   E*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)