Theorem sb8mo 1989

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 1811	. . 3 |- ( E. x ph <-> E. y [ y / x ] ph )
3	1	sb8eu 1988	. . 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 1979	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6		df-mo 1979	. 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 1419   E.wex 1451   [wsb 1718   E!weu 1975   E*wmo 1976

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979

This theorem is referenced by: (None)