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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-wnfanf 36701 | When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023.) |
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)) | ||
Theorem | bj-wnfenf 36702 | When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the existential form of nonfreeness. (Contributed by BJ, 9-Dec-2023.) |
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → 𝜓)) | ||
Theorem | bj-substax12 36703 |
Equivalent form of the axiom of substitution bj-ax12 36639. Although both
sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 36667 on
𝑡,
𝜑) to hold, their
equivalence holds without DV conditions. The
forward implication is proved in modal (K4) while the reverse implication
is proved in modal (T5). The LHS has the advantage of not involving
nested quantifiers on the same variable. Its metaweakening is proved from
the core axiom schemes in bj-substw 36704. Note that in the LHS, the reverse
implication holds by equs4 2418 (or equs4v 1996 if a DV condition is added on
𝑥,
𝑡 as in bj-ax12 36639), and the forward implication is sbalex 2239.
The LHS can be read as saying that if there exists a variable equal to a given term witnessing a given formula, then all variables equal to that term also witness that formula. The equivalent form of the LHS using only primitive symbols is (∀𝑥(𝑥 = 𝑡 → 𝜑) ∨ ∀𝑥(𝑥 = 𝑡 → ¬ 𝜑)), which expresses that a given formula is true at all variables equal to a given term, or false at all these variables. An equivalent form of the LHS using only the existential quantifier is ¬ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑡 ∧ ¬ 𝜑)), which expresses that there can be no two variables both equal to a given term, one witnessing a formula and the other witnessing its negation. These equivalences do not hold in intuitionistic logic. The LHS should be the preferred form, and has the advantage of having no negation nor nested quantifiers. (Contributed by BJ, 21-May-2024.) (Proof modification is discouraged.) |
⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))) | ||
Theorem | bj-substw 36704* | Weak form of the LHS of bj-substax12 36703 proved from the core axiom schemes. Compare ax12w 2130. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) | ||
Syntax | wnnf 36705 | Syntax for the nonfreeness quantifier. |
wff Ⅎ'𝑥𝜑 | ||
Definition | df-bj-nnf 36706 |
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 1907 and ax5e 1909. 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 36736, 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 2156 and excom 2159 (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 1907, ax5e 1909, ax5ea 1910. 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 36707 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 36736, 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 36728, bj-nnfnt 36722, bj-nnfan 36730, bj-nnfor 36732, bj-nnfbit 36734, bj-nnfalt 36748, bj-nnfext 36749. 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 36728 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 36748 yields ((∀𝑦∀𝑥1 ...∀𝑥m Ⅎ'𝑥𝜑 → Ⅎ'𝑥∀𝑦 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 36748 and bj-nnfext 36749 are proved from positive propositional calculus with alcom 2156 and excom 2159 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 36743). 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 1907 or ax5e 1909 or ax5ea 1910. 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 36710 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 1909 or ax5ea 1910, we would use bj-nnfe 36713 or bj-nnfea 36716 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 1791, by bj-nnf-alrim 36737 and bj-nnfa1 36741 (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 1977. QED Compared with df-nf 1780, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 36720) and equivalent on core FOL plus sp 2180 (bj-nfnnfTEMP 36740). While being stricter, it still holds for non-occurring variables (bj-nnfv 36736), 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 ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑)) ∧ ∀𝑥∀𝑥... 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.) |
⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) | ||
Theorem | bj-nnfbi 36707 | 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 1847. From this and bj-nnfim 36728 and bj-nnfnt 36722, 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 36723) in order not to require sp 2180 (modal T). (Contributed by BJ, 27-Aug-2023.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)) | ||
Theorem | bj-nnfbd 36708* | If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, deduction form. See bj-nnfbi 36707. (Contributed by BJ, 27-Aug-2023.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒)) | ||
Theorem | bj-nnfbii 36709 | If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 36707. (Contributed by BJ, 18-Nov-2023.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓) | ||
Theorem | bj-nnfa 36710 | Nonfreeness implies the equivalent of ax-5 1907. See nf5r 2191. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nnfad 36711 | Nonfreeness implies the equivalent of ax-5 1907, deduction form. See nf5rd 2193. (Contributed by BJ, 2-Dec-2023.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | bj-nnfai 36712 | Nonfreeness implies the equivalent of ax-5 1907, inference form. See nf5ri 2192. (Contributed by BJ, 22-Sep-2024.) |
⊢ Ⅎ'𝑥𝜑 ⇒ ⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | bj-nnfe 36713 | Nonfreeness implies the equivalent of ax5e 1909. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) | ||
Theorem | bj-nnfed 36714 | Nonfreeness implies the equivalent of ax5e 1909, deduction form. (Contributed by BJ, 2-Dec-2023.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓)) | ||
Theorem | bj-nnfei 36715 | Nonfreeness implies the equivalent of ax5e 1909, inference form. (Contributed by BJ, 22-Sep-2024.) |
⊢ Ⅎ'𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 → 𝜑) | ||
Theorem | bj-nnfea 36716 | Nonfreeness implies the equivalent of ax5ea 1910. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nnfead 36717 | Nonfreeness implies the equivalent of ax5ea 1910, deduction form. (Contributed by BJ, 2-Dec-2023.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | ||
Theorem | bj-nnfeai 36718 | Nonfreeness implies the equivalent of ax5ea 1910, inference form. (Contributed by BJ, 22-Sep-2024.) |
⊢ Ⅎ'𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 → ∀𝑥𝜑) | ||
Theorem | bj-dfnnf2 36719 | Alternate definition of df-bj-nnf 36706 using only primitive symbols (→, ¬, ∀) in each conjunct. (Contributed by BJ, 20-Aug-2023.) |
⊢ (Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))) | ||
Theorem | bj-nnfnfTEMP 36720 | New nonfreeness implies old nonfreeness on minimal implicational calculus (the proof indicates it uses ax-3 8 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1780 except via df-nf 1780 directly. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑) | ||
Theorem | bj-wnfnf 36721 | When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 36728, bj-nnfe1 36742 and bj-nnfa1 36741. (Contributed by BJ, 9-Dec-2023.) |
⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | bj-nnfnt 36722 | A variable is nonfree in a formula if and only if it is nonfree in its negation. The foward implication is intuitionistically valid (and that direction is sufficient for the purpose of recursively proving that some formulas have a given variable not free in them, like bj-nnfim 36728). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1853. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑) | ||
Theorem | bj-nnftht 36723 | A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2180 (modal T), as in bj-nnfbi 36707. (Contributed by BJ, 28-Jul-2023.) |
⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑) | ||
Theorem | bj-nnfth 36724 | A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.) |
⊢ 𝜑 ⇒ ⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnfnth 36725 | A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.) |
⊢ ¬ 𝜑 ⇒ ⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnfim1 36726 | A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((𝜑 → 𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-nnfim2 36727 | A consequence of nonfreeness in the antecedent and the consequent of an implication. (Contributed by BJ, 27-Aug-2023.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → ((∀𝑥𝜑 → ∃𝑥𝜓) → (𝜑 → 𝜓))) | ||
Theorem | bj-nnfim 36728 | Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication. (Contributed by BJ, 27-Aug-2023.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 → 𝜓)) | ||
Theorem | bj-nnfimd 36729 | Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 → 𝜒)) | ||
Theorem | bj-nnfan 36730 | Nonfreeness in both conjuncts implies nonfreeness in the conjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of conjunction in terms of implication and negation, so using bj-nnfim 36728, bj-nnfnt 36722 and bj-nnfbi 36707, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | bj-nnfand 36731 | Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 36730, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 36730 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 36731 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∧ 𝜒)) | ||
Theorem | bj-nnfor 36732 | Nonfreeness in both disjuncts implies nonfreeness in the disjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of disjunction in terms of implication and negation, so using bj-nnfim 36728, bj-nnfnt 36722 and bj-nnfbi 36707, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | bj-nnford 36733 | Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 36732 and bj-nnfand 36731. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒)) | ||
Theorem | bj-nnfbit 36734 | Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ↔ 𝜓)) | ||
Theorem | bj-nnfbid 36735 | Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ↔ 𝜒)) | ||
Theorem | bj-nnfv 36736* | A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.) |
⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnf-alrim 36737 | Proof of the closed form of alrimi 2210 from modalK (compare alrimiv 1924). See also bj-alrim 36675. Actually, most proofs between 19.3t 2198 and 2sbbid 2244 could be proved without ax-12 2174. (Contributed by BJ, 20-Aug-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-nnf-exlim 36738 | Proof of the closed form of exlimi 2214 from modalK (compare exlimiv 1927). See also bj-sylget2 36604. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))) | ||
Theorem | bj-dfnnf3 36739 | Alternate definition of nonfreeness when sp 2180 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1780. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nfnnfTEMP 36740 | New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2180. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1780 except via df-nf 1780 directly. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑) | ||
Theorem | bj-nnfa1 36741 | See nfa1 2148. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ Ⅎ'𝑥∀𝑥𝜑 | ||
Theorem | bj-nnfe1 36742 | See nfe1 2147. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ Ⅎ'𝑥∃𝑥𝜑 | ||
Theorem | bj-19.12 36743 | See 19.12 2325. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2159 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1780 or df-bj-nnf 36706, directly or indirectly. (Proof modification is discouraged.) |
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) | ||
Theorem | bj-nnflemaa 36744 | One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 36663. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | ||
Theorem | bj-nnflemee 36745 | One of four lemmas for nonfreeness: antecedent and consequent both expressed using existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑)) | ||
Theorem | bj-nnflemae 36746 | One of four lemmas for nonfreeness: antecedent expressed with universal quantifier and consequent expressed with existential quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∃𝑥𝜑 → ∀𝑦∃𝑥𝜑)) | ||
Theorem | bj-nnflemea 36747 | One of four lemmas for nonfreeness: antecedent expressed with existential quantifier and consequent expressed with universal quantifier. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(∃𝑦𝜑 → 𝜑) → (∃𝑦∀𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nnfalt 36748 | See nfal 2321 and bj-nfalt 36693. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑) | ||
Theorem | bj-nnfext 36749 | See nfex 2322 and bj-nfext 36694. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑) | ||
Theorem | bj-stdpc5t 36750 | Alias of bj-nnf-alrim 36737 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2205 proved from modalK (obsoleting stdpc5v 1935). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 36737 instead. (New usaged is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-19.21t 36751 | Statement 19.21t 2203 proved from modalK (obsoleting 19.21v 1936). (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-19.23t 36752 | Statement 19.23t 2207 proved from modalK (obsoleting 19.23v 1939). (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
Theorem | bj-19.36im 36753 | One direction of 19.36 2227 from the same axioms as 19.36imv 1942. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))) | ||
Theorem | bj-19.37im 36754 | One direction of 19.37 2229 from the same axioms as 19.37imv 1944. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))) | ||
Theorem | bj-19.42t 36755 | Closed form of 19.42 2233 from the same axioms as 19.42v 1950. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))) | ||
Theorem | bj-19.41t 36756 | Closed form of 19.41 2232 from the same axioms as 19.41v 1946. The same is doable with 19.27 2224, 19.28 2225, 19.31 2231, 19.32 2230, 19.44 2234, 19.45 2235. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))) | ||
Theorem | bj-sbft 36757 | Version of sbft 2267 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.) |
⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | bj-pm11.53vw 36758 | Version of pm11.53v 1941 with nonfreeness antecedents. One can also prove the theorem with antecedent (Ⅎ'𝑦∀𝑥𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓). (Contributed by BJ, 7-Oct-2024.) |
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-pm11.53v 36759 | Version of pm11.53v 1941 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.) |
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-pm11.53a 36760* | A variant of pm11.53v 1941. One can similarly prove a variant with DV (𝑦, 𝜑) and ∀𝑦Ⅎ'𝑥𝜓 instead of DV (𝑥, 𝜓) and ∀𝑥Ⅎ'𝑦𝜑. (Contributed by BJ, 7-Oct-2024.) |
⊢ (∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-equsvt 36761* | A variant of equsv 1999. (Contributed by BJ, 7-Oct-2024.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) | ||
Theorem | bj-equsalvwd 36762* | Variant of equsalvw 2000. (Contributed by BJ, 7-Oct-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒)) | ||
Theorem | bj-equsexvwd 36763* | Variant of equsexvw 2001. (Contributed by BJ, 7-Oct-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒)) | ||
Theorem | bj-sbievwd 36764* | Variant of sbievw 2090. (Contributed by BJ, 7-Oct-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | bj-axc10 36765 | Alternate proof of axc10 2387. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2374, by using ax6ev 1966 instead of ax6e 2385. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | bj-alequex 36766 | A fol lemma. See alequexv 1997 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2388 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | ||
Theorem | bj-spimt2 36767 | A step in the proof of spimt 2388. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓))) | ||
Theorem | bj-cbv3ta 36768 | Closed form of cbv3 2399. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-cbv3tb 36769 | Closed form of cbv3 2399. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-hbsb3t 36770 | A theorem close to a closed form of hbsb3 2489. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) | ||
Theorem | bj-hbsb3 36771 | Shorter proof of hbsb3 2489. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1t 36772 | A theorem close to a closed form of nfs1 2490. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1t2 36773 | A theorem close to a closed form of nfs1 2490. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1 36774 | Shorter proof of nfs1 2490 (three essential steps instead of four). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
It is known that ax-13 2374 is logically redundant (see ax13w 2133 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2374 from every theorem in set.mm which is totally unbundled (i.e., has disjoint variable conditions on all setvar variables). Indeed, start with the existing proof, and replace any occurrence of ax-13 2374 with ax13w 2133. This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2374 (and using ax6v 1965 / ax6ev 1966 instead of ax-6 1964 / ax6e 2385, as is currently done). One reason to be optimistic is that the first few utility theorems using ax-13 2374 (roughly 200 of them) are then used mainly with dummy variables, which one can assume distinct from any other, so that the unbundled versions of the utility theorems suffice. In this section, we prove versions of theorems in the main part with dv conditions and not requiring ax-13 2374, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1965 and ax6ev 1966 instead of ax-6 1964 and ax6e 2385, and ax-5 1907 instead of ax13v 2375; shorter proofs may be possible). When no additional dv condition is required, we label it bj-xxx. It is important to keep all the bundled theorems already in set.mm, but one may also add the (partially) unbundled versions which dispense with ax-13 2374, so as to remove dependencies on ax-13 2374 from many existing theorems. UPDATE: it turns out that several theorems of the form bj-xxxv, or minor variations, are already in set.mm with label xxxw. It is also possible to remove dependencies on ax-11 2154, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2238 and following theorems). | ||
Theorem | bj-axc10v 36775* | Version of axc10 2387 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | bj-spimtv 36776* | Version of spimt 2388 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-cbv3hv2 36777* | Version of cbv3h 2406 with two disjoint variable conditions, which does not require ax-11 2154 nor ax-13 2374. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | bj-cbv1hv 36778* | Version of cbv1h 2407 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) | ||
Theorem | bj-cbv2hv 36779* | Version of cbv2h 2408 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbv2v 36780* | Version of cbv2 2405 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbvaldv 36781* | Version of cbvald 2409 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbvexdv 36782* | Version of cbvexd 2410 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | bj-cbval2vv 36783* | Version of cbval2vv 2415 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) | ||
Theorem | bj-cbvex2vv 36784* | Version of cbvex2vv 2416 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) | ||
Theorem | bj-cbvaldvav 36785* | Version of cbvaldva 2411 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbvexdvav 36786* | Version of cbvexdva 2412 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | bj-cbvex4vv 36787* | Version of cbvex4v 2417 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓)) & ⊢ ((𝑧 = 𝑓 ∧ 𝑤 = 𝑔) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑣∃𝑢∃𝑓∃𝑔𝜒) | ||
Theorem | bj-equsalhv 36788* |
Version of equsalh 2422 with a disjoint variable condition, which
does not
require ax-13 2374. Remark: this is the same as equsalhw 2289. TODO:
delete after moving the following paragraph somewhere.
Remarks: equsexvw 2001 has been moved to Main; Theorem ax13lem2 2378 has a DV version which is a simple consequence of ax5e 1909; Theorems nfeqf2 2379, dveeq2 2380, nfeqf1 2381, dveeq1 2382, nfeqf 2383, axc9 2384, ax13 2377, have dv versions which are simple consequences of ax-5 1907. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | bj-axc11nv 36789* | Version of axc11n 2428 with a disjoint variable condition; instance of aevlem 2052. TODO: delete after checking surrounding theorems. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | bj-aecomsv 36790* | Version of aecoms 2430 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2431 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5446). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | bj-axc11v 36791* | Version of axc11 2432 with a disjoint variable condition, which does not require ax-13 2374 nor ax-10 2138. Remark: the following theorems (hbae 2433, nfae 2435, hbnae 2434, nfnae 2436, hbnaes 2437) would need to be totally unbundled to be proved without ax-13 2374, hence would be simple consequences of ax-5 1907 or nfv 1911. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Theorem | bj-drnf2v 36792* | Version of drnf2 2446 with a disjoint variable condition, which does not require ax-10 2138, ax-11 2154, ax-12 2174, ax-13 2374. Instance of nfbidv 1919. Note that the version of axc15 2424 with a disjoint variable condition is actually ax12v2 2176 (up to adding a superfluous antecedent). (Contributed by BJ, 17-Jun-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓)) | ||
Theorem | bj-equs45fv 36793* | Version of equs45f 2461 with a disjoint variable condition, which does not require ax-13 2374. Note that the version of equs5 2462 with a disjoint variable condition is actually sbalex 2239 (up to adding a superfluous antecedent). (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | bj-hbs1 36794* | Version of hbsb2 2484 with a disjoint variable condition, which does not require ax-13 2374, and removal of ax-13 2374 from hbs1 2271. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1v 36795* | Version of nfsb2 2485 with a disjoint variable condition, which does not require ax-13 2374, and removal of ax-13 2374 from nfs1v 2153. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
Theorem | bj-hbsb2av 36796* | Version of hbsb2a 2486 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-hbsb3v 36797* | Version of hbsb3 2489 with a disjoint variable condition, which does not require ax-13 2374. (Remark: the unbundled version of nfs1 2490 is given by bj-nfs1v 36795.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfsab1 36798* | Remove dependency on ax-13 2374 from nfsab1 2719. UPDATE / TODO: nfsab1 2719 does not use ax-13 2374 either anymore; bj-nfsab1 36798 is shorter than nfsab1 2719 but uses ax-12 2174. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | bj-dtrucor2v 36799* | Version of dtrucor2 5377 with a disjoint variable condition, which does not require ax-13 2374 (nor ax-4 1805, ax-5 1907, ax-7 2004, ax-12 2174). (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → 𝑥 ≠ 𝑦) ⇒ ⊢ (𝜑 ∧ ¬ 𝜑) | ||
The closed formula ∀𝑥∀𝑦𝑥 = 𝑦 approximately means that the var metavariables 𝑥 and 𝑦 represent the same variable vi. In a domain with at most one object, however, this formula is always true, hence the "approximately" in the previous sentence. | ||
Theorem | bj-hbaeb2 36800 | Biconditional version of a form of hbae 2433 with commuted quantifiers, not requiring ax-11 2154. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦) |
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