Theorem sb8mo 2504

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 2425	. . 3 |- ( E. x ph <-> E. y [ y / x ] ph )
3	1	sb8eu 2503	. . 3 |- ( E! x ph <-> E! y [ y / x ] ph )
4	2, 3	imbi12i 340	. 2 |- ( ( E. x ph -> E! x ph ) <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) )
5		df-mo 2475	. 2 |- ( E* x ph <-> ( E. x ph -> E! x ph ) )
6		df-mo 2475	. 2 |- ( E* y [ y / x ] ph <-> ( E. y [ y / x ] ph -> E! y [ y / x ] ph ) )
7	4, 5, 6	3bitr4i 292	1 |- ( E* x ph <-> E* y [ y / x ] ph )

Syntax hints:   -> wi 4   <-> wb 196   E.wex 1704   F/wnf 1708   [wsb 1880   E!weu 2470   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)