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Metamath Proof Explorer

Theorem elOLD

Description: Obsolete version of el as of 6-Apr-2026. (Contributed by NM, 4-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion elOLD 
|- E. y x e. y

Proof

Step Hyp Ref Expression
1 ax-pr 
 |-  E. y A. z ( ( z = x \/ z = x ) -> z e. y )
2 pm4.25 
 |-  ( z = x <-> ( z = x \/ z = x ) )
3 2 imbi1i 
 |-  ( ( z = x -> z e. y ) <-> ( ( z = x \/ z = x ) -> z e. y ) )
4 3 albii 
 |-  ( A. z ( z = x -> z e. y ) <-> A. z ( ( z = x \/ z = x ) -> z e. y ) )
5 elequ1 
 |-  ( z = x -> ( z e. y <-> x e. y ) )
6 5 equsalvw 
 |-  ( A. z ( z = x -> z e. y ) <-> x e. y )
7 4 6 bitr3i 
 |-  ( A. z ( ( z = x \/ z = x ) -> z e. y ) <-> x e. y )
8 7 exbii 
 |-  ( E. y A. z ( ( z = x \/ z = x ) -> z e. y ) <-> E. y x e. y )
9 1 8 mpbi 
 |-  E. y x e. y