Theorem sb8mo 1955

Description: Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
sb8eu.1	|- F/ y ph

Assertion

Ref	Expression
sb8mo	|- ( E* x ph <-> E* y [ y / x ] ph )

Proof of Theorem sb8mo

Step	Hyp	Ref	Expression
1		sb8eu.1	. . . 4 |- F/ y ph
2	1	sb8e 1778	. . 3 |- ( E. x ph <-> E. y [ y / x ] ph )
3	1	sb8eu 1954	. . 3 |- ( E! x ph <-> E! y [ y / x ] ph )
4	2, 3	imbi12i 237	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) )
5		df-mo 1945	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6		df-mo 1945	. 2 |- ( E* y [ y / x ] ph <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) )
7	4, 5, 6	3bitr4i 210	1 |- ( E* x ph <-> E* y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   E.wex 1421   [wsb 1685   E!weu 1941   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-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by: (None)