Metamath Proof Explorer


Theorem bj-vtoclg

Description: A version of vtoclg with an additional disjoint variable condition (which is removable if we allow use of df-clab , see bj-vtoclg1f ), which requires fewer axioms (i.e., removes dependency on ax-6 , ax-7 , ax-9 , ax-12 , ax-ext , df-clab , df-cleq , df-v ). (Contributed by BJ, 2-Jul-2022) (Proof modification is discouraged.)

Ref Expression
Hypotheses bj-vtoclg.maj
|- ( x = A -> ( ph -> ps ) )
bj-vtoclg.min
|- ph
Assertion bj-vtoclg
|- ( A e. V -> ps )

Proof

Step Hyp Ref Expression
1 bj-vtoclg.maj
 |-  ( x = A -> ( ph -> ps ) )
2 bj-vtoclg.min
 |-  ph
3 elissetv
 |-  ( A e. V -> E. x x = A )
4 1 2 bj-exlimvmpi
 |-  ( E. x x = A -> ps )
5 3 4 syl
 |-  ( A e. V -> ps )