 Thus weak extensionality is congruentiality ‘with side formulas’ (collected into Γ). The obvious adaptation of this notion to generalized consequence relations would be to allow side-formulas on the right too (replacing the above“⊢” by “⊩” and adding a set variable “Δ” to the right-hand sides, that is): this turns out to be equivalent to extensionality for generalized consequence relations as defined above in terms of (LEi) and (REi). ⊢I L is weakly extensional in this sense, and weak extensionality according to ⊢ ⊇ ⊢I L is a property of some theoretical significance for the case in which the language of ⊢ adds new connectives in the spirit of the original intuitionistic  primitives.[10] 