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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-wnfanf 35901 | 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 35902 | 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 35903 |
Equivalent form of the axiom of substitution bj-ax12 35838. Although both
sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 35867 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 35904. Note that in the LHS, the reverse
implication holds by equs4 2414 (or equs4v 2002 if a DV condition is added on
𝑥,
𝑡 as in bj-ax12 35838), and the forward implication is sbalex 2234.
The LHS can be read as saying that if there exists a setvar equal to a given term witnessing 𝜑, then all setvars equal to that term also witness 𝜑. An equivalent suggestive form for the LHS 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 𝜑 and the other witnessing ¬ 𝜑. (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 35904* | Weak form of the LHS of bj-substax12 35903 proved from the core axiom schemes. Compare ax12w 2128. (Contributed by BJ, 26-May-2024.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) | ||
Syntax | wnnf 35905 | Syntax for the nonfreeness quantifier. |
wff Ⅎ'𝑥𝜑 | ||
Definition | df-bj-nnf 35906 |
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 1912 and ax5e 1914. 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 35936, 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 2155 and excom 2161 (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 1912, ax5e 1914, ax5ea 1915. 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 35907 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 35936, 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 35928, bj-nnfnt 35922, bj-nnfan 35930, bj-nnfor 35932, bj-nnfbit 35934, bj-nnfalt 35948, bj-nnfext 35949. 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 35928 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 35948 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 35948 and bj-nnfext 35949 are proved from positive propositional calculus with alcom 2155 and excom 2161 (possibly weakened by a DV condition on the quantifying variables), and modalB (via bj-19.12 35943). 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 1912 or ax5e 1914 or ax5ea 1915. 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 35910 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 1914 or ax5ea 1915, we would use bj-nnfe 35913 or bj-nnfea 35916 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 1796, by bj-nnf-alrim 35937 and bj-nnfa1 35941 (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 1983. QED Compared with df-nf 1785, the present definition is stricter on positive propositional calculus (bj-nnfnfTEMP 35920) and equivalent on core FOL plus sp 2175 (bj-nfnnfTEMP 35940). While being stricter, it still holds for non-occurring variables (bj-nnfv 35936), 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 2136 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 35907 | 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 1852. From this and bj-nnfim 35928 and bj-nnfnt 35922, 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 35923) in order not to require sp 2175 (modal T). (Contributed by BJ, 27-Aug-2023.) |
⊢ (((𝜑 ↔ 𝜓) ∧ ∀𝑥(𝜑 ↔ 𝜓)) → (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)) | ||
Theorem | bj-nnfbd 35908* | 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 35907. (Contributed by BJ, 27-Aug-2023.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (Ⅎ'𝑥𝜓 ↔ Ⅎ'𝑥𝜒)) | ||
Theorem | bj-nnfbii 35909 | 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 35907. (Contributed by BJ, 18-Nov-2023.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓) | ||
Theorem | bj-nnfa 35910 | Nonfreeness implies the equivalent of ax-5 1912. See nf5r 2186. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nnfad 35911 | Nonfreeness implies the equivalent of ax-5 1912, deduction form. See nf5rd 2188. (Contributed by BJ, 2-Dec-2023.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | bj-nnfai 35912 | Nonfreeness implies the equivalent of ax-5 1912, inference form. See nf5ri 2187. (Contributed by BJ, 22-Sep-2024.) |
⊢ Ⅎ'𝑥𝜑 ⇒ ⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | bj-nnfe 35913 | Nonfreeness implies the equivalent of ax5e 1914. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) | ||
Theorem | bj-nnfed 35914 | Nonfreeness implies the equivalent of ax5e 1914, deduction form. (Contributed by BJ, 2-Dec-2023.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜓)) | ||
Theorem | bj-nnfei 35915 | Nonfreeness implies the equivalent of ax5e 1914, inference form. (Contributed by BJ, 22-Sep-2024.) |
⊢ Ⅎ'𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 → 𝜑) | ||
Theorem | bj-nnfea 35916 | Nonfreeness implies the equivalent of ax5ea 1915. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nnfead 35917 | Nonfreeness implies the equivalent of ax5ea 1915, deduction form. (Contributed by BJ, 2-Dec-2023.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓)) | ||
Theorem | bj-nnfeai 35918 | Nonfreeness implies the equivalent of ax5ea 1915, inference form. (Contributed by BJ, 22-Sep-2024.) |
⊢ Ⅎ'𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 → ∀𝑥𝜑) | ||
Theorem | bj-dfnnf2 35919 | Alternate definition of df-bj-nnf 35906 using only primitive symbols (→, ¬, ∀) in each conjunct. (Contributed by BJ, 20-Aug-2023.) |
⊢ (Ⅎ'𝑥𝜑 ↔ ((𝜑 → ∀𝑥𝜑) ∧ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))) | ||
Theorem | bj-nnfnfTEMP 35920 | 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 1785 except via df-nf 1785 directly. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑) | ||
Theorem | bj-wnfnf 35921 | 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 35928, bj-nnfe1 35942 and bj-nnfa1 35941. (Contributed by BJ, 9-Dec-2023.) |
⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | bj-nnfnt 35922 | 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 35928). Intuitionistically, ⊢ (Ⅎ'𝑥¬ 𝜑 ↔ Ⅎ'𝑥¬ ¬ 𝜑). See nfnt 1858. (Contributed by BJ, 28-Jul-2023.) |
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥 ¬ 𝜑) | ||
Theorem | bj-nnftht 35923 | A variable is nonfree in a theorem. The antecedent is in the "strong necessity" modality of modal logic in order not to require sp 2175 (modal T), as in bj-nnfbi 35907. (Contributed by BJ, 28-Jul-2023.) |
⊢ ((𝜑 ∧ ∀𝑥𝜑) → Ⅎ'𝑥𝜑) | ||
Theorem | bj-nnfth 35924 | A variable is nonfree in a theorem, inference form. (Contributed by BJ, 28-Jul-2023.) |
⊢ 𝜑 ⇒ ⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnfnth 35925 | A variable is nonfree in the negation of a theorem, inference form. (Contributed by BJ, 27-Aug-2023.) |
⊢ ¬ 𝜑 ⇒ ⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnfim1 35926 | 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 35927 | 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 35928 | 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 35929 | 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 35930 | 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 35928, bj-nnfnt 35922 and bj-nnfbi 35907, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | bj-nnfand 35931 | Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 35930, 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 35930 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 35931 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 35932 | 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 35928, bj-nnfnt 35922 and bj-nnfbi 35907, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.) |
⊢ ((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | bj-nnford 35933 | Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 35932 and bj-nnfand 35931. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.) |
⊢ (𝜑 → Ⅎ'𝑥𝜓) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ'𝑥(𝜓 ∨ 𝜒)) | ||
Theorem | bj-nnfbit 35934 | 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 35935 | 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 35936* | A non-occurring variable is nonfree in a formula. (Contributed by BJ, 28-Jul-2023.) |
⊢ Ⅎ'𝑥𝜑 | ||
Theorem | bj-nnf-alrim 35937 | Proof of the closed form of alrimi 2205 from modalK (compare alrimiv 1929). See also bj-alrim 35875. Actually, most proofs between 19.3t 2193 and 2sbbid 2238 could be proved without ax-12 2170. (Contributed by BJ, 20-Aug-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-nnf-exlim 35938 | Proof of the closed form of exlimi 2209 from modalK (compare exlimiv 1932). See also bj-sylget2 35803. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))) | ||
Theorem | bj-dfnnf3 35939 | Alternate definition of nonfreeness when sp 2175 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1785. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | bj-nfnnfTEMP 35940 | New nonfreeness is equivalent to old nonfreeness on core FOL axioms plus sp 2175. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1785 except via df-nf 1785 directly. (Proof modification is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 ↔ Ⅎ𝑥𝜑) | ||
Theorem | bj-nnfa1 35941 | See nfa1 2147. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ Ⅎ'𝑥∀𝑥𝜑 | ||
Theorem | bj-nnfe1 35942 | See nfe1 2146. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ Ⅎ'𝑥∃𝑥𝜑 | ||
Theorem | bj-19.12 35943 | See 19.12 2319. Could be labeled "exalimalex" for "'there exists for all' implies 'for all there exists'". This proof is from excom 2161 and modal (B) on top of modalK logic. (Contributed by BJ, 12-Aug-2023.) The proof should not rely on df-nf 1785 or df-bj-nnf 35906, directly or indirectly. (Proof modification is discouraged.) |
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) | ||
Theorem | bj-nnflemaa 35944 | One of four lemmas for nonfreeness: antecedent and consequent both expressed using universal quantifier. Note: this is bj-hbalt 35863. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)) | ||
Theorem | bj-nnflemee 35945 | 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 35946 | 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 35947 | 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 35948 | See nfal 2315 and bj-nfalt 35893. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∀𝑥𝜑) | ||
Theorem | bj-nnfext 35949 | See nfex 2316 and bj-nfext 35894. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥Ⅎ'𝑦𝜑 → Ⅎ'𝑦∃𝑥𝜑) | ||
Theorem | bj-stdpc5t 35950 | Alias of bj-nnf-alrim 35937 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2200 proved from modalK (obsoleting stdpc5v 1940). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 35937 instead. (New usaged is discouraged.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-19.21t 35951 | Statement 19.21t 2198 proved from modalK (obsoleting 19.21v 1941). (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-19.23t 35952 | Statement 19.23t 2202 proved from modalK (obsoleting 19.23v 1944). (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
Theorem | bj-19.36im 35953 | One direction of 19.36 2222 from the same axioms as 19.36imv 1947. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))) | ||
Theorem | bj-19.37im 35954 | One direction of 19.37 2224 from the same axioms as 19.37imv 1950. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))) | ||
Theorem | bj-19.42t 35955 | Closed form of 19.42 2228 from the same axioms as 19.42v 1956. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))) | ||
Theorem | bj-19.41t 35956 | Closed form of 19.41 2227 from the same axioms as 19.41v 1952. The same is doable with 19.27 2219, 19.28 2220, 19.31 2226, 19.32 2225, 19.44 2229, 19.45 2230. (Contributed by BJ, 2-Dec-2023.) |
⊢ (Ⅎ'𝑥𝜓 → (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))) | ||
Theorem | bj-sbft 35957 | Version of sbft 2260 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.) |
⊢ (Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | bj-pm11.53vw 35958 | Version of pm11.53v 1946 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 35959 | Version of pm11.53v 1946 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.) |
⊢ ((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-pm11.53a 35960* | A variant of pm11.53v 1946. 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 35961* | A variant of equsv 2005. (Contributed by BJ, 7-Oct-2024.) |
⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) | ||
Theorem | bj-equsalvwd 35962* | Variant of equsalvw 2006. (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 35963* | Variant of equsexvw 2007. (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 35964* | Variant of sbievw 2094. (Contributed by BJ, 7-Oct-2024.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → Ⅎ'𝑥𝜒) & ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | bj-axc10 35965 | Alternate proof of axc10 2383. Shorter. One can prove a version with DV (𝑥, 𝑦) without ax-13 2370, by using ax6ev 1972 instead of ax6e 2381. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | bj-alequex 35966 | A fol lemma. See alequexv 2003 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2384 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | ||
Theorem | bj-spimt2 35967 | A step in the proof of spimt 2384. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓))) | ||
Theorem | bj-cbv3ta 35968 | Closed form of cbv3 2395. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-cbv3tb 35969 | Closed form of cbv3 2395. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-hbsb3t 35970 | A theorem close to a closed form of hbsb3 2485. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) | ||
Theorem | bj-hbsb3 35971 | Shorter proof of hbsb3 2485. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1t 35972 | A theorem close to a closed form of nfs1 2486. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1t2 35973 | A theorem close to a closed form of nfs1 2486. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1 35974 | Shorter proof of nfs1 2486 (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 2370 is logically redundant (see ax13w 2131 and the head comment of the section "Logical redundancy of ax-10--13"). More precisely, one can remove dependency on ax-13 2370 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 2370 with ax13w 2131. This section is an experiment to see in practice if (partially) unbundled versions of existing theorems can be proved more efficiently without ax-13 2370 (and using ax6v 1971 / ax6ev 1972 instead of ax-6 1970 / ax6e 2381, as is currently done). One reason to be optimistic is that the first few utility theorems using ax-13 2370 (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 2370, labeled bj-xxxv (we follow the proof of xxx but use ax6v 1971 and ax6ev 1972 instead of ax-6 1970 and ax6e 2381, and ax-5 1912 instead of ax13v 2371; 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 dipense with ax-13 2370, so as to remove dependencies on ax-13 2370 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 2153, typically by replacing a nonfree hypothesis with a disjoint variable condition (see cbv3v2 2233 and following theorems). | ||
Theorem | bj-axc10v 35975* | Version of axc10 2383 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | bj-spimtv 35976* | Version of spimt 2384 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 14-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-cbv3hv2 35977* | Version of cbv3h 2402 with two disjoint variable conditions, which does not require ax-11 2153 nor ax-13 2370. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | bj-cbv1hv 35978* | Version of cbv1h 2403 with a disjoint variable condition, which does not require ax-13 2370. (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 35979* | Version of cbv2h 2404 with a disjoint variable condition, which does not require ax-13 2370. (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 35980* | Version of cbv2 2401 with a disjoint variable condition, which does not require ax-13 2370. (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 35981* | Version of cbvald 2405 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbvexdv 35982* | Version of cbvexd 2406 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | bj-cbval2vv 35983* | Version of cbval2vv 2411 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) | ||
Theorem | bj-cbvex2vv 35984* | Version of cbvex2vv 2412 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) | ||
Theorem | bj-cbvaldvav 35985* | Version of cbvaldva 2407 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | bj-cbvexdvav 35986* | Version of cbvexdva 2408 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | bj-cbvex4vv 35987* | Version of cbvex4v 2413 with a disjoint variable condition, which does not require ax-13 2370. (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 35988* |
Version of equsalh 2418 with a disjoint variable condition, which
does not
require ax-13 2370. Remark: this is the same as equsalhw 2286. TODO:
delete after moving the following paragraph somewhere.
Remarks: equsexvw 2007 has been moved to Main; Theorem ax13lem2 2374 has a DV version which is a simple consequence of ax5e 1914; Theorems nfeqf2 2375, dveeq2 2376, nfeqf1 2377, dveeq1 2378, nfeqf 2379, axc9 2380, ax13 2373, have dv versions which are simple consequences of ax-5 1912. (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 35989* | Version of axc11n 2424 with a disjoint variable condition; instance of aevlem 2057. 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 35990* | Version of aecoms 2426 with a disjoint variable condition, provable from Tarski's FOL. The corresponding version of naecoms 2427 should not be very useful since ¬ ∀𝑥𝑥 = 𝑦, DV (𝑥, 𝑦) is true when the universe has at least two objects (see dtru 5437). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | bj-axc11v 35991* | Version of axc11 2428 with a disjoint variable condition, which does not require ax-13 2370 nor ax-10 2136. Remark: the following theorems (hbae 2429, nfae 2431, hbnae 2430, nfnae 2432, hbnaes 2433) would need to be totally unbundled to be proved without ax-13 2370, hence would be simple consequences of ax-5 1912 or nfv 1916. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Theorem | bj-drnf2v 35992* | Version of drnf2 2442 with a disjoint variable condition, which does not require ax-10 2136, ax-11 2153, ax-12 2170, ax-13 2370. Instance of nfbidv 1924. Note that the version of axc15 2420 with a disjoint variable condition is actually ax12v2 2172 (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 35993* | Version of equs45f 2457 with a disjoint variable condition, which does not require ax-13 2370. Note that the version of equs5 2458 with a disjoint variable condition is actually sbalex 2234 (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 35994* | Version of hbsb2 2480 with a disjoint variable condition, which does not require ax-13 2370, and removal of ax-13 2370 from hbs1 2264. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfs1v 35995* | Version of nfsb2 2481 with a disjoint variable condition, which does not require ax-13 2370, and removal of ax-13 2370 from nfs1v 2152. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
Theorem | bj-hbsb2av 35996* | Version of hbsb2a 2482 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-hbsb3v 35997* | Version of hbsb3 2485 with a disjoint variable condition, which does not require ax-13 2370. (Remark: the unbundled version of nfs1 2486 is given by bj-nfs1v 35995.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | bj-nfsab1 35998* | Remove dependency on ax-13 2370 from nfsab1 2716. UPDATE / TODO: nfsab1 2716 does not use ax-13 2370 either anymore; bj-nfsab1 35998 is shorter than nfsab1 2716 but uses ax-12 2170. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) |
⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | ||
Theorem | bj-dtrucor2v 35999* | Version of dtrucor2 5371 with a disjoint variable condition, which does not require ax-13 2370 (nor ax-4 1810, ax-5 1912, ax-7 2010, ax-12 2170). (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 36000 | Biconditional version of a form of hbae 2429 with commuted quantifiers, not requiring ax-11 2153. (Contributed by BJ, 12-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥∀𝑧 𝑥 = 𝑦) |
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