Theorem bj-evaleq

Description: Equality theorem for the Slot construction. This is currently a duplicate of sloteq but may diverge from it if/when a token Eval is introduced for evaluation in order to separate it from Slot and any of its possible modifications. (Contributed by BJ, 27-Dec-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-evaleq	⊢ ( 𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝐴 = 𝐵 → ( 𝑓 ‘ 𝐴 ) = ( 𝑓 ‘ 𝐵 ) )
2	1	mpteq2dv	⊢ ( 𝐴 = 𝐵 → ( 𝑓 ∈ V ↦ ( 𝑓 ‘ 𝐴 ) ) = ( 𝑓 ∈ V ↦ ( 𝑓 ‘ 𝐵 ) ) )
3		df-slot	⊢ Slot 𝐴 = ( 𝑓 ∈ V ↦ ( 𝑓 ‘ 𝐴 ) )
4		df-slot	⊢ Slot 𝐵 = ( 𝑓 ∈ V ↦ ( 𝑓 ‘ 𝐵 ) )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵 )