Theorem rmo2 3526

Description: Alternate definition of restricted "at most one." Note that E* x e. A ph is not equivalent to E. y e. A A. x e. A ( ph -> x = y ) (in analogy to reu6 3395); to see this, let A be the empty set. However, one direction of this pattern holds; see rmo2i 3527. (Contributed by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	|- F/ y ph

Assertion

Ref	Expression
rmo2	|- ( E* x e. A ph <-> E. y A. x e. A ( ph -> x = y ) )

Distinct variable group:   x, y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem rmo2

Step	Hyp	Ref	Expression
1		df-rmo 2920	. 2 |- ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
2		nfv 1843	. . . 4 |- F/ y x e. A
3		rmo2.1	. . . 4 |- F/ y ph
4	2, 3	nfan 1828	. . 3 |- F/ y ( x e. A /\ ph )
5	4	mo2 2479	. 2 |- ( E* x ( x e. A /\ ph ) <-> E. y A. x ( ( x e. A /\ ph ) -> x = y ) )
6		impexp 462	. . . . 5 |- ( ( ( x e. A /\ ph ) -> x = y ) <-> ( x e. A -> ( ph -> x = y ) ) )
7	6	albii 1747	. . . 4 |- ( A. x ( ( x e. A /\ ph ) -> x = y ) <-> A. x ( x e. A -> ( ph -> x = y ) ) )
8		df-ral 2917	. . . 4 |- ( A. x e. A ( ph -> x = y ) <-> A. x ( x e. A -> ( ph -> x = y ) ) )
9	7, 8	bitr4i 267	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> x = y ) <-> A. x e. A ( ph -> x = y ) )
10	9	exbii 1774	. 2 |- ( E. y A. x ( ( x e. A /\ ph ) -> x = y ) <-> E. y A. x e. A ( ph -> x = y ) )
11	1, 5, 10	3bitri 286	1 |- ( E* x e. A ph <-> E. y A. x e. A ( ph -> x = y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708   e. wcel 1990   E*wmo 2471   A.wral 2912   E*wrmo 2915

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-eu 2474  df-mo 2475  df-ral 2917  df-rmo 2920

This theorem is referenced by:  rmo2i  3527  disjiun  4640  rmoeqALT  29327  poimirlem2  33411  rmoanim  41179