Theorem rabeqsn 4001

Description: Conditions for a restricted class abstraction to be a singleton. (Contributed by AV, 18-Apr-2019.)

Assertion

Ref	Expression
rabeqsn	|- ( { x e. V | ph } = { X } <-> A. x ( ( x e. V /\ ph ) <-> x = X ) )

Distinct variable group:   x, X

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem rabeqsn

Step	Hyp	Ref	Expression
1		df-rab 2746	. . 3 |- { x e. V | ph } = { x | ( x e. V /\ ph ) }
2		df-sn 3969	. . 3 |- { X } = { x | x = X }
3	1, 2	eqeq12i 2465	. 2 |- ( { x e. V | ph } = { X } <-> { x | ( x e. V /\ ph ) } = { x | x = X } )
4		abbi 2565	. 2 |- ( A. x ( ( x e. V /\ ph ) <-> x = X ) <-> { x | ( x e. V /\ ph ) } = { x | x = X } )
5	3, 4	bitr4i 256	1 |- ( { x e. V | ph } = { X } <-> A. x ( ( x e. V /\ ph ) <-> x = X ) )

Syntax hints:   <-> wb 188   /\ wa 371   A.wal 1442   = wceq 1444   e. wcel 1887   {cab 2437   {crab 2741   {csn 3968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-rab 2746  df-sn 3969

This theorem is referenced by:  umgr2v2enb1  39563