Metamath Proof Explorer

Theorem bj-inex1gALT

Description: Proof of inex1g from sepg to then allow proving inex1 from it. That does not reduce the combined proof size of inex1 and inex1g . (Contributed by BJ, 14-Jul-2026) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion bj-inex1gALT 
|- ( A e. V -> ( A i^i B ) e. _V )

Proof

Step Hyp Ref Expression
1 sepg 
 |-  ( A e. V -> E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) ) )
2 dfcleq 
 |-  ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
3 elin 
 |-  ( y e. ( A i^i B ) <-> ( y e. A /\ y e. B ) )
4 3 a1i 
 |-  ( A e. V -> ( y e. ( A i^i B ) <-> ( y e. A /\ y e. B ) ) )
5 4 bibi2d 
 |-  ( A e. V -> ( ( y e. x <-> y e. ( A i^i B ) ) <-> ( y e. x <-> ( y e. A /\ y e. B ) ) ) )
6 5 albidv 
 |-  ( A e. V -> ( A. y ( y e. x <-> y e. ( A i^i B ) ) <-> A. y ( y e. x <-> ( y e. A /\ y e. B ) ) ) )
7 2 6 bitrid 
 |-  ( A e. V -> ( x = ( A i^i B ) <-> A. y ( y e. x <-> ( y e. A /\ y e. B ) ) ) )
8 7 exbidv 
 |-  ( A e. V -> ( E. x x = ( A i^i B ) <-> E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) ) ) )
9 1 8 mpbird 
 |-  ( A e. V -> E. x x = ( A i^i B ) )
10 isset 
 |-  ( ( A i^i B ) e. _V <-> E. x x = ( A i^i B ) )
11 9 10 sylibr 
 |-  ( A e. V -> ( A i^i B ) e. _V )