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Metamath Proof Explorer

Theorem bj-nnfbi

Description: If two formulas are equivalent for all x , then nonfreeness of x in one of them is equivalent to nonfreeness in the other. Compare nfbiit . From this and bj-nnfim and bj-nnfnt , one can prove analogous nonfreeness conservation results for other propositional operators. The antecedent is in the "strong necessity" modality of modal logic (see also bj-nnftht ) in order not to require sp (modal T). (Contributed by BJ, 27-Aug-2023)

Ref Expression
Assertion bj-nnfbi 
|- ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( F// x ph <-> F// x ps ) )

Proof

Step Hyp Ref Expression
1 bj-hbyfrbi 
 |-  ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( ( E. x ph -> ph ) <-> ( E. x ps -> ps ) ) )
2 bj-hbxfrbi 
 |-  ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( ( ph -> A. x ph ) <-> ( ps -> A. x ps ) ) )
3 1 2 anbi12d 
 |-  ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( ( ( E. x ph -> ph ) /\ ( ph -> A. x ph ) ) <-> ( ( E. x ps -> ps ) /\ ( ps -> A. x ps ) ) ) )
4 df-bj-nnf 
 |-  ( F// x ph <-> ( ( E. x ph -> ph ) /\ ( ph -> A. x ph ) ) )
5 df-bj-nnf 
 |-  ( F// x ps <-> ( ( E. x ps -> ps ) /\ ( ps -> A. x ps ) ) )
6 3 4 5 3bitr4g 
 |-  ( ( ( ph <-> ps ) /\ A. x ( ph <-> ps ) ) -> ( F// x ph <-> F// x ps ) )