Theorem sb8mo 2011

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 1829	. . 3 |- ( E. x ph <-> E. y [ y / x ] ph )
3	1	sb8eu 2010	. . 3 |- ( E! x ph <-> E! y [ y / x ] ph )
4	2, 3	imbi12i 238	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) )
5		df-mo 2001	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6		df-mo 2001	. 2 |- ( E* y [ y / x ] ph <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) )
7	4, 5, 6	3bitr4i 211	1 |- ( E* x ph <-> E* y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 104   F/wnf 1436   E.wex 1468   [wsb 1735   E!weu 1997   E*wmo 1998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001

This theorem is referenced by: (None)