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Definition df-bj-nnf 35218
Description: Definition of the nonfreeness quantifier. The formula Ⅎ'𝑥𝜑 has the intended meaning that the variable 𝑥 is semantically nonfree in the formula 𝜑. The motivation for this quantifier is to have a condition expressible in the logic which is as close as possible to the non-occurrence condition DV (𝑥, 𝜑) (in Metamath files, "$d x ph $."), which belongs to the metalogic.

The standard syntactic nonfreeness condition, also expressed in the metalogic, is intermediate between these two notions: semantic nonfreeness implies syntactic nonfreeness, which implies non-occurrence. Both implications are strict; for the first, note that Ⅎ'𝑥𝑥 = 𝑥, that is, 𝑥 is semantically (but not syntactically) nonfree in the formula 𝑥 = 𝑥; for the second, note that 𝑥 is syntactically nonfree in the formula 𝑥𝑥 = 𝑥 although it occurs in it.

We now prove two metatheorems which make precise the above fact that, as far as proving power is concerned, the nonfreeness condition Ⅎ'𝑥𝜑 is very close to the non-occurrence condition DV (𝑥, 𝜑).

Let S be a Metamath system with the FOL-syntax of (i)set.mm, containing intuitionistic positive propositional calculus and ax-5 1914 and ax5e 1916.

Theorem 1. If the scheme

(Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn PHI0, DV)

is provable in S, then so is the scheme

(PHI1 & ... & PHIn PHI0, DV ∪ {{𝑥, 𝜑}}).

Proof: By bj-nnfv 35248, we can prove (Ⅎ'𝑥𝜑, {{𝑥, 𝜑}}), from which the theorem follows. QED

Theorem 2. Suppose that S also contains (the FOL version of) modal logic KB and commutation of quantifiers alcom 2157 and excom 2163 (possibly weakened by a DV condition on the quantifying variables), and that S can be axiomatized such that the only axioms with a DV condition involving a formula variable are among ax-5 1914, ax5e 1916, ax5ea 1917. If the scheme

(PHI1 & ... & PHIn PHI0, DV)

is provable in S, then so is the scheme

(Ⅎ'𝑥𝜑 & PHI1 & ... & PHIn PHI0, DV ∖ {{𝑥, 𝜑}}).

More precisely, if S contains modal 45 and if the variables quantified over in PHI0, ..., PHIn are among 𝑥1, ..., 𝑥m, then the scheme

(PHI1 & ... & PHIn (antecedent PHI0), DV ∖ {{𝑥, 𝜑}})

is provable in S, where the antecedent is a finite conjunction of formulas of the form 𝑥i1 ...∀𝑥ip Ⅎ'𝑥𝜑 where the 𝑥ij's are among the 𝑥i's.

Lemma: If 𝑥 OC(PHI), then S proves the scheme

(Ⅎ'𝑥𝜑 ⇒ Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}).

More precisely, if the variables quantified over in PHI are among 𝑥1, ..., 𝑥m, then

((antecedent → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}})

is provable in S, with the same form of antecedent as above.

Proof: By induction on the height of PHI. We first note that by bj-nnfbi 35219 we can assume that PHI contains only primitive (as opposed to defined) symbols. For the base case, atomic formulas are either 𝜑, in which case the scheme to prove is an instance of id 22, or have variables all in OC(PHI) ∖ {𝜑}, so (Ⅎ'𝑥 PHI, {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}) by bj-nnfv 35248, hence ((Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}) by a1i 11. For the induction step, PHI is either an implication, a negation, a conjunction, a disjunction, a biconditional, a universal or an existential quantification of formulas where 𝑥 does not occur. We use respectively bj-nnfim 35240, bj-nnfnt 35234, bj-nnfan 35242, bj-nnfor 35244, bj-nnfbit 35246, bj-nnfalt 35260, bj-nnfext 35261. For instance, in the implication case, if we have by induction hypothesis

((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}) and ((∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 OC(PSI) ∖ {𝜑}}),

then bj-nnfim 35240 yields

(((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 ∧ ∀𝑦1 ...∀𝑦n Ⅎ'𝑥𝜑) → Ⅎ'𝑥 (PHI PSI)), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI PSI) ∖ {𝜑}})

and similarly for antecedents which are conjunctions as in the statement of the lemma.

In the universal quantification case, say quantification over 𝑦, if we have by induction hypothesis

((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(PHI) ∖ {𝜑}}),

then bj-nnfalt 35260 yields

((∀𝑦𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥𝑦 PHI), {{𝑥, 𝑎} ∣ 𝑎 OC(𝑦 PHI) ∖ {𝜑}})

and similarly for antecedents which are conjunctions as in the statement of the lemma.

Note bj-nnfalt 35260 and bj-nnfext 35261 are proved from positive propositional calculus with alcom 2157 and excom 2163 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 35255). QED

Proof of the theorem: Consider a proof of that scheme directly from the axioms. Consider a step where a DV condition involving 𝜑 is used. By hypothesis, that step is an instance of ax-5 1914 or ax5e 1916 or ax5ea 1917. It has the form (PSI → ∀𝑥 PSI) where PSI has the form of the lemma and the DV conditions of the proof contain {{𝑥, 𝑎} ∣ 𝑎 OC(PSI) }. Therefore, one has

((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥 PSI), {{𝑥, 𝑎} ∣ 𝑎 OC(PSI) ∖ {𝜑}})

for appropriate 𝑥i's, and by bj-nnfa 35222 we obtain

((∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 (PSI → ∀𝑥 PSI)), {{𝑥, 𝑎} ∣ 𝑎 OC(PSI) ∖ {𝜑}})

and similarly for antecedents which are conjunctions as in the statement of the theorem. Similarly if the step is using ax5e 1916 or ax5ea 1917, we would use bj-nnfe 35225 or bj-nnfea 35228 respectively.

Therefore, taking as antecedent of the theorem to prove the conjunction of all the antecedents at each of these steps, we obtain a proof by "carrying the context over", which is possible, as in the deduction theorem when the step uses ax-mp 5, and when the step uses ax-gen 1798, by bj-nnf-alrim 35249 and bj-nnfa1 35253 (which requires modal 45). The condition DV (𝑥, 𝜑) is not required by the resulting proof.

Finally, there may be in the global antecedent thus constructed some dummy variables, which can be removed by spvw 1985. QED

Compared with df-nf 1787, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 35232) and equivalent on core FOL plus sp 2177 (bj-nfnnfTEMP 35252). While being stricter, it still holds for non-occurring variables (bj-nnfv 35248), which is the basic requirement for this quantifier. In particular, it translates more closely the associated variable disjointness condition. Since the nonfreeness quantifier is a means to translate a variable disjointness condition from the metalogic to the logic, it seems preferable. Also, since nonfreeness is mainly used as a hypothesis, this definition would allow more theorems, notably the 19.xx theorems, to be proved from the core axioms, without needing a 19.xxv variant.

One can devise infinitely many definitions increasingly close to the non-occurring condition, like ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ 𝑥((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥𝑥... and each stronger definition would permit more theorems to be proved from the core axioms. A reasonable rule seems to be to stop before nested quantifiers appear (since they typically require ax-10 2138 to work with), and also not to have redundant conjuncts when full metacomplete FOL= is developed.

(Contributed by BJ, 28-Jul-2023.)

Assertion
Ref Expression
df-bj-nnf (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))

Detailed syntax breakdown of Definition df-bj-nnf
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 vx . . 3 setvar 𝑥
31, 2wnnf 35217 . 2 wff Ⅎ'𝑥𝜑
41, 2wex 1782 . . . 4 wff 𝑥𝜑
54, 1wi 4 . . 3 wff (∃𝑥𝜑𝜑)
61, 2wal 1540 . . . 4 wff 𝑥𝜑
71, 6wi 4 . . 3 wff (𝜑 → ∀𝑥𝜑)
85, 7wa 397 . 2 wff ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑))
93, 8wb 205 1 wff (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑𝜑) ∧ (𝜑 → ∀𝑥𝜑)))
Colors of variables: wff setvar class
This definition is referenced by:  bj-nnfbi  35219  bj-nnfa  35222  bj-nnfe  35225  bj-dfnnf2  35231  bj-wnfnf  35233  bj-nnfnt  35234  bj-nnftht  35235  bj-nnfim  35240  bj-nnfan  35242  bj-nnfand  35243  bj-nnfor  35244  bj-nnford  35245  bj-nnfv  35248  bj-dfnnf3  35251  bj-nnfa1  35253  bj-nnfe1  35254  bj-nnfalt  35260  bj-nnfext  35261
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