Theorem sb8mo 2069

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 1881	. . 3 |- ( E. x ph <-> E. y [ y / x ] ph )
3	1	sb8eu 2068	. . 3 |- ( E! x ph <-> E! y [ y / x ] ph )
4	2, 3	imbi12i 239	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) )
5		df-mo 2059	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6		df-mo 2059	. 2 |- ( E* y [ y / x ] ph <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) )
7	4, 5, 6	3bitr4i 212	1 |- ( E* x ph <-> E* y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1484   E.wex 1516   [wsb 1786   E!weu 2055   E*wmo 2056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059

This theorem is referenced by: (None)