Theorem rmoeqALT 29327

Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) Obsolete version of rmoeq 3405 as of 27-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
rmoeqALT	|- ( A e. V -> E* x e. B x = A )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   V( x)

Proof of Theorem rmoeqALT

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( x = A -> x = A )
2	1	rgenw 2924	. . 3 |- A. x e. B ( x = A -> x = A )
3		eqeq2 2633	. . . . . 6 |- ( y = A -> ( x = y <-> x = A ) )
4	3	imbi2d 330	. . . . 5 |- ( y = A -> ( ( x = A -> x = y ) <-> ( x = A -> x = A ) ) )
5	4	ralbidv 2986	. . . 4 |- ( y = A -> ( A. x e. B ( x = A -> x = y ) <-> A. x e. B ( x = A -> x = A ) ) )
6	5	spcegv 3294	. . 3 |- ( A e. V -> ( A. x e. B ( x = A -> x = A ) -> E. y A. x e. B ( x = A -> x = y ) ) )
7	2, 6	mpi 20	. 2 |- ( A e. V -> E. y A. x e. B ( x = A -> x = y ) )
8		nfv 1843	. . 3 |- F/ y x = A
9	8	rmo2 3526	. 2 |- ( E* x e. B x = A <-> E. y A. x e. B ( x = A -> x = y ) )
10	7, 9	sylibr 224	1 |- ( A e. V -> E* x e. B x = A )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rmo 2920  df-v 3202

This theorem is referenced by: (None)