Theorem prv

Description: The "proves" relation on a set. A wff encoded as U is true in a model M iff for every valuation s e. ( M ^m _om ) , the interpretation of the wff using the membership relation on M is true. (Contributed by AV, 5-Nov-2023)

	Ref	Expression
Assertion	prv	|- ( ( M e. V /\ U e. W ) -> ( M |= U <-> ( M SatE U ) = ( M ^m _om ) ) )

Proof

Step	Hyp	Ref	Expression
1		oveq12	|- ( ( m = M /\ u = U ) -> ( m SatE u ) = ( M SatE U ) )
2		simpl	|- ( ( m = M /\ u = U ) -> m = M )
3	2	oveq1d	|- ( ( m = M /\ u = U ) -> ( m ^m _om ) = ( M ^m _om ) )
4	1 3	eqeq12d	|- ( ( m = M /\ u = U ) -> ( ( m SatE u ) = ( m ^m _om ) <-> ( M SatE U ) = ( M ^m _om ) ) )
5		df-prv	|- |= = { <. m , u >. | ( m SatE u ) = ( m ^m _om ) }
6	4 5	brabga	|- ( ( M e. V /\ U e. W ) -> ( M |= U <-> ( M SatE U ) = ( M ^m _om ) ) )