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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bj-cbvexim 33201* | A lemma used to prove justification theorems in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (∀𝑥∃𝑦𝜒 → (∀𝑥∀𝑦(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜑 → ∃𝑦𝜓))) | ||
Theorem | bj-cbvalimi 33202* | An equality-free general instance of one half of a precise form of bj-cbval 33204. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜒 → (𝜑 → 𝜓)) & ⊢ ∀𝑦∃𝑥𝜒 ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | bj-cbveximi 33203* | An equality-free general instance of one half of a precise form of bj-cbvex 33205. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ (𝜒 → (𝜑 → 𝜓)) & ⊢ ∀𝑥∃𝑦𝜒 ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-cbval 33204* | Changing a bound variable (universal quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦 & ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | bj-cbvex 33205* | Changing a bound variable (existential quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
⊢ ∀𝑦∃𝑥 𝑥 = 𝑦 & ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Syntax | wmoo 33206 | Syntax for BJ's version of the uniqueness quantifier. |
wff ∃**𝑥𝜑 | ||
Definition | df-bj-mo 33207* | Definition of the uniqueness quantifier which is correct on the empty domain. Instead of the fresh variable 𝑧, one could save a dummy variable by using 𝑥 or 𝑦 at the cost of having nested quantifiers on the same variable. (Contributed by BJ, 12-Mar-2023.) |
⊢ (∃**𝑥𝜑 ↔ ∀𝑧∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) | ||
Theorem | bj-ssbjustlem 33208* | Lemma for bj-ssbjust 33209. (Contributed by BJ, 9-Nov-2021.) |
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | ||
Theorem | bj-ssbjust 33209* | Justification theorem for df-ssb 33211 from Tarski's FOL. (Contributed by BJ, 22-Jan-2023.) |
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | ||
Syntax | wssb 33210 | Syntax for the substitution of a variable for a variable in a formula. (Contributed by BJ, 22-Dec-2020.) |
wff [𝑡/𝑥]b𝜑 | ||
Definition | df-ssb 33211* |
Alternate definition of proper substitution. Note that the occurrences
of a given variable are either all bound (𝑥, 𝑦) or all free
(𝑡). Also note that the definiens uses
only primitive symbols.
It is obtained by applying twice Tarski's definition sb6 2250
which is
valid for disjoint variables, so we introduce a dummy variable 𝑦 to
isolate 𝑥 from 𝑡, as in dfsb7 2535 with respect to sb5 2251.
This double level definition will make several proofs using it appear as doubled. Alternately, one could often first prove as a lemma the same theorem with a disjoint variable condition on the substitute and the substituted variables, and then prove the original theorem by applying this lemma twice in a row. This definition uses a dummy variable, but the justification theorem, bj-ssbjust 33209, is provable from Tarski's FOL. Once this is proved, more of the fundamental properties of proper substitution will be provable from Tarski's FOL system, sometimes with the help of specialization sp 2167, of the substitution axiom ax-12 2163, and of commutation of quantifiers ax-11 2150; that is, ax-13 2334 will often be avoided. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-ssbim 33212 | Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2470. (Contributed by BJ, 22-Dec-2020.) |
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓)) | ||
Theorem | bj-ssbbi 33213 | Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2478. (Contributed by BJ, 22-Dec-2020.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓)) | ||
Theorem | bj-ssbimi 33214 | Distribute substitution over implication. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓) | ||
Theorem | bj-ssbbii 33215 | Biconditional property for substitution. Uses only ax-1--5. (Contributed by BJ, 22-Dec-2020.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓) | ||
Theorem | bj-alsb 33216 | If a proposition is true for all instances, then it is true for any specific one. Uses only ax-1--5. Compare stdpc4 2428 which uses auxiliary axioms. (Contributed by BJ, 22-Dec-2020.) |
⊢ (∀𝑥𝜑 → [𝑡/𝑥]b𝜑) | ||
Theorem | bj-sbex 33217 | If a proposition is true for a specific instance, then there exists an instance such that it is true for it. Uses only ax-1--6. Compare spsbe 2015 which, due to the specific form of df-sb 2012, uses fewer axioms. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑡/𝑥]b𝜑 → ∃𝑥𝜑) | ||
Theorem | bj-ssbeq 33218* | Substitution in an equality, disjoint variables case. Uses only ax-1--6. It might be shorter to prove the result about composition of two substitutions and prove bj-ssbeq 33218 first with a DV on x,t, and then in the general case. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑡/𝑥]b𝑦 = 𝑧 ↔ 𝑦 = 𝑧) | ||
Theorem | bj-ssb0 33219* | Substitution for a variable not occurring in a proposition. See sbf 2456. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑡/𝑥]b𝜑 ↔ 𝜑) | ||
Theorem | bj-ssbequ 33220 | Equality property for substitution, from Tarski's system. Compare sbequ 2452. (Contributed by BJ, 30-Dec-2020.) |
⊢ (𝑠 = 𝑡 → ([𝑠/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑)) | ||
Theorem | bj-ssblem1 33221* | A lemma for the definiens of df-sb 2012. An instance of sp 2167 proved without it. Note: it has a common subproof with bj-ssbjust 33209. (Contributed by BJ, 22-Dec-2020.) |
⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-ssblem2 33222* | An instance of ax-11 2150 proved without it. The converse may not be provable without ax-11 2150 (since using alcomiw 2088 would require a DV on 𝜑, 𝑥, which defeats the purpose). (Contributed by BJ, 22-Dec-2020.) |
⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-ssb1a 33223* | One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 33224 for the biconditional, which requires ax-11 2150. (Contributed by BJ, 22-Dec-2020.) |
⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → [𝑡/𝑥]b𝜑) | ||
Theorem | bj-ssb1 33224* | A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 33223 for the backward implication, which does not require ax-11 2150 (note that here, the version of ax-11 2150 with disjoint setvar metavariables would suffice). Compare sb6 2250. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) | ||
Theorem | bj-ax12 33225* | A weaker form of ax-12 2163 and ax12v2 2165, namely the generalization over 𝑥 of the latter. In this statement, all occurrences of 𝑥 are bound. (Contributed by BJ, 26-Dec-2020.) |
⊢ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))) | ||
Theorem | bj-ax12ssb 33226* | The axiom bj-ax12 33225 expressed using substitution. (Contributed by BJ, 26-Dec-2020.) |
⊢ [𝑡/𝑥]b(𝜑 → [𝑡/𝑥]b𝜑) | ||
Theorem | bj-19.41al 33227 | Special case of 19.41 2221 proved from Tarski, ax-10 2135 (modal5) and hba1 2268 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | bj-equsexval 33228* | Special case of equsexv 2242 proved from Tarski, ax-10 2135 (modal5) and hba1 2268 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥𝜓) | ||
Theorem | bj-sb56 33229* | Proof of sb56 2248 from Tarski, ax-10 2135 (modal5) and bj-ax12 33225. (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | bj-dfssb2 33230* | An alternate definition of df-ssb 33211. Note that the use of a dummy variable in the definition df-ssb 33211 allows to use bj-sb56 33229 instead of equs45f 2425 and hence to avoid dependency on ax-13 2334 and to use ax-12 2163 only through bj-ax12 33225. Compare dfsb7 2535. (Contributed by BJ, 25-Dec-2020.) |
⊢ ([𝑡/𝑥]b𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | ||
Theorem | bj-ssbn 33231 | The result of a substitution in the negation of a formula is the negation of the result of the same substitution in that formula. Proved from Tarski, ax-10 2135, bj-ax12 33225. Compare sbn 2467. (Contributed by BJ, 25-Dec-2020.) |
⊢ ([𝑡/𝑥]b ¬ 𝜑 ↔ ¬ [𝑡/𝑥]b𝜑) | ||
Theorem | bj-ssbft 33232 | See sbft 2455. This proof is from Tarski's FOL together with sp 2167 (and its dual). (Contributed by BJ, 22-Dec-2020.) |
⊢ (Ⅎ𝑥𝜑 → ([𝑡/𝑥]b𝜑 ↔ 𝜑)) | ||
Theorem | bj-ssbequ2 33233 | Note that ax-12 2163 is used only via sp 2167. See sbequ2 2013 and stdpc7 2076. (Contributed by BJ, 22-Dec-2020.) |
⊢ (𝑥 = 𝑡 → ([𝑡/𝑥]b𝜑 → 𝜑)) | ||
Theorem | bj-ssbequ1 33234 | This uses ax-12 2163 with a direct reference to ax12v 2164. Therefore, compared to bj-ax12 33225, there is a hidden use of sp 2167. Note that with ax-12 2163, it can be proved with disjoint variable condition on 𝑥, 𝑡. See sbequ1 2228. (Contributed by BJ, 22-Dec-2020.) |
⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡/𝑥]b𝜑)) | ||
Theorem | bj-ssbid2 33235 | A special case of bj-ssbequ2 33233. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑥/𝑥]b𝜑 → 𝜑) | ||
Theorem | bj-ssbid2ALT 33236 | Alternate proof of bj-ssbid2 33235, not using bj-ssbequ2 33233. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ([𝑥/𝑥]b𝜑 → 𝜑) | ||
Theorem | bj-ssbid1 33237 | A special case of bj-ssbequ1 33234. (Contributed by BJ, 22-Dec-2020.) |
⊢ (𝜑 → [𝑥/𝑥]b𝜑) | ||
Theorem | bj-ssbid1ALT 33238 | Alternate proof of bj-ssbid1 33237, not using bj-ssbequ1 33234. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → [𝑥/𝑥]b𝜑) | ||
Theorem | bj-ssbssblem 33239* | Composition of two substitutions with a fresh intermediate variable. Remark: does not seem useful. (Contributed by BJ, 22-Dec-2020.) |
⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜑) | ||
Theorem | bj-ssbcom3lem 33240* | Lemma for bj-ssbcom3 when setvar variables are disjoint. Remark: does not seem useful. (Contributed by BJ, 30-Dec-2020.) |
⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑦]b[𝑡/𝑥]b𝜑) | ||
Theorem | bj-ax6elem1 33241* | Lemma for bj-ax6e 33243. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
Theorem | bj-ax6elem2 33242* | Lemma for bj-ax6e 33243. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑦 = 𝑧 → ∃𝑥 𝑥 = 𝑦) | ||
Theorem | bj-ax6e 33243 | Proof of ax6e 2347 (hence ax6 2348) from Tarski's system, ax-c9 35046, ax-c16 35048. Remark: ax-6 2021 is used only via its principal (unbundled) instance ax6v 2022. (Contributed by BJ, 22-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃𝑥 𝑥 = 𝑦 | ||
Theorem | bj-alequexv 33244* | Version of bj-alequex 33296 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | ||
Theorem | bj-spimvwt 33245* | Closed form of spimvw 2045. See also spimt 2350. (Contributed by BJ, 8-Nov-2021.) |
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-spimevw 33246* | Existential introduction, using implicit substitution. This is to spimeh 2043 what spimvw 2045 is to spimw 2044. (Contributed by BJ, 17-Mar-2020.) |
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | bj-spnfw 33247 | Theorem close to a closed form of spnfw 2046. (Contributed by BJ, 12-May-2019.) |
⊢ ((∃𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-cbvexiw 33248* | Change bound variable. This is to cbvexvw 2087 what cbvaliw 2053 is to cbvalvw 2086. [TODO: move after cbvalivw 2054]. (Contributed by BJ, 17-Mar-2020.) |
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-cbvexivw 33249* | Change bound variable. This is to cbvexvw 2087 what cbvalivw 2054 is to cbvalvw 2086. [TODO: move after cbvalivw 2054]. (Contributed by BJ, 17-Mar-2020.) |
⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓)) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑦𝜓) | ||
Theorem | bj-modald 33250 | A short form of the axiom D of modal logic. (Contributed by BJ, 4-Apr-2021.) |
⊢ (∀𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) | ||
Theorem | bj-denot 33251* | A weakening of ax-6 2021 and ax6v 2022. (Contributed by BJ, 4-Apr-2021.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑥 → ¬ ∀𝑦 ¬ 𝑦 = 𝑥) | ||
Theorem | bj-eqs 33252* | A lemma for substitutions, proved from Tarski's FOL. The version without DV(x,y) is true but requires ax-13 2334. The disjoint variable condition DV(x,ph) is necessary for both directions: consider substituting 𝑥 = 𝑧 for 𝜑. (Contributed by BJ, 25-May-2021.) |
⊢ (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | bj-cbvexw 33253* | Change bound variable. This is to cbvexvw 2087 what cbvalw 2085 is to cbvalvw 2086. (Contributed by BJ, 17-Mar-2020.) |
⊢ (∃𝑥∃𝑦𝜓 → ∃𝑦𝜓) & ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (∃𝑦∃𝑥𝜑 → ∃𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Theorem | bj-ax12w 33254* | The general statement that ax12w 2127 proves. (Contributed by BJ, 20-Mar-2020.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ (𝑦 = 𝑧 → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (∀𝑦𝜓 → ∀𝑥(𝜑 → 𝜓))) | ||
Theorem | bj-elequ2g 33255* | A form of elequ2 2121 with a universal quantifier. Its converse is ax-ext 2754. (TODO: move to main part, minimize axext4 2758--- as of 4-Nov-2020, minimizes only axext4 2758, by 13 bytes; and link to it in the comment of ax-ext 2754.) (Contributed by BJ, 3-Oct-2019.) |
⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)) | ||
Theorem | bj-ax89 33256 | A theorem which could be used as sole axiom for the non-logical predicate instead of ax-8 2109 and ax-9 2116. Indeed, it is implied over propositional calculus by the conjunction of ax-8 2109 and ax-9 2116, as proved here. In the other direction, one can prove ax-8 2109 (respectively ax-9 2116) from bj-ax89 33256 by using mpan2 681 ( respectively mpan 680) and equid 2059. (TODO: move to main part.) (Contributed by BJ, 3-Oct-2019.) |
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑡)) | ||
Theorem | bj-elequ12 33257 | An identity law for the non-logical predicate, which combines elequ1 2114 and elequ2 2121. For the analogous theorems for class terms, see eleq1 2847, eleq2 2848 and eleq12 2849. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.) |
⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑡)) | ||
Theorem | bj-cleljusti 33258* | One direction of cleljust 2115, requiring only ax-1 6-- ax-5 1953 and ax8v1 2111. (Contributed by BJ, 31-Dec-2020.) (Proof modification is discouraged.) |
⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) → 𝑥 ∈ 𝑦) | ||
Theorem | bj-alcomexcom 33259 | Commutation of universal quantifiers implies commutation of existential quantifiers. Can be placed in the ax-4 1853 section, soon after 2nexaln 1873, and used to prove excom 2155. (Contributed by BJ, 29-Nov-2020.) (Proof modification is discouraged.) |
⊢ ((∀𝑥∀𝑦 ¬ 𝜑 → ∀𝑦∀𝑥 ¬ 𝜑) → (∃𝑦∃𝑥𝜑 → ∃𝑥∃𝑦𝜑)) | ||
Theorem | bj-hbalt 33260 | Closed form of hbal 2160. When in main part, prove hbal 2160 and hbald 2161 from it. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑)) | ||
Theorem | axc11n11 33261 | Proof of axc11n 2392 from { ax-1 6-- ax-7 2055, axc11 2396 } . Almost identical to axc11nfromc11 35082. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | axc11n11r 33262 |
Proof of axc11n 2392 from { ax-1 6--
ax-7 2055, axc9 2346, axc11r 2333 } (note
that axc16 2234 is provable from { ax-1 6--
ax-7 2055, axc11r 2333 }).
Note that axc11n 2392 proves (over minimal calculus) that axc11 2396 and axc11r 2333 are equivalent. Therefore, axc11n11 33261 and axc11n11r 33262 prove that one can use one or the other as an axiom, provided one assumes the axioms listed above (axc11 2396 appears slightly stronger since axc11n11r 33262 requires axc9 2346 while axc11n11 33261 does not). (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | bj-axc16g16 33263* | Proof of axc16g 2233 from { ax-1 6-- ax-7 2055, axc16 2234 }. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | ||
Theorem | bj-ax12v3 33264* | A weak version of ax-12 2163 which is stronger than ax12v 2164. Note that if one assumes reflexivity of equality ⊢ 𝑥 = 𝑥 (equid 2059), then bj-ax12v3 33264 implies ax-5 1953 over modal logic K (substitute 𝑥 for 𝑦). See also bj-ax12v3ALT 33265. (Contributed by BJ, 6-Jul-2021.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-ax12v3ALT 33265* | Alternate proof of bj-ax12v3 33264. Uses axc11r 2333 and axc15 2387 instead of ax-12 2163. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-sb 33266* | A weak variant of sbid2 2489 not requiring ax-13 2334 nor ax-10 2135. On top of Tarski's FOL, one implication requires only ax12v 2164, and the other requires only sp 2167. (Contributed by BJ, 25-May-2021.) |
⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | bj-modalbe 33267 | The predicate-calculus version of the axiom (B) of modal logic. See also modal-b 2294. (Contributed by BJ, 20-Oct-2019.) |
⊢ (𝜑 → ∀𝑥∃𝑥𝜑) | ||
Theorem | bj-spst 33268 | Closed form of sps 2169. Once in main part, prove sps 2169 and spsd 2171 from it. (Contributed by BJ, 20-Oct-2019.) |
⊢ ((𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | bj-19.21bit 33269 | Closed form of 19.21bi 2173. (Contributed by BJ, 20-Oct-2019.) |
⊢ ((𝜑 → ∀𝑥𝜓) → (𝜑 → 𝜓)) | ||
Theorem | bj-19.23bit 33270 | Closed form of 19.23bi 2175. (Contributed by BJ, 20-Oct-2019.) |
⊢ ((∃𝑥𝜑 → 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | bj-nexrt 33271 | Closed form of nexr 2176. Contrapositive of 19.8a 2166. (Contributed by BJ, 20-Oct-2019.) |
⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑) | ||
Theorem | bj-alrim 33272 | Closed form of alrimi 2199. (Contributed by BJ, 2-May-2019.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | ||
Theorem | bj-alrim2 33273 | Uncurried (imported) form of bj-alrim 33272. (Contributed by BJ, 2-May-2019.) |
⊢ ((Ⅎ𝑥𝜑 ∧ ∀𝑥(𝜑 → 𝜓)) → (𝜑 → ∀𝑥𝜓)) | ||
Theorem | bj-nfdt0 33274 | A theorem close to a closed form of nf5d 2258 and nf5dh 2141. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → (∀𝑥𝜑 → Ⅎ𝑥𝜓)) | ||
Theorem | bj-nfdt 33275 | Closed form of nf5d 2258 and nf5dh 2141. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → (𝜓 → ∀𝑥𝜓)) → ((𝜑 → ∀𝑥𝜑) → (𝜑 → Ⅎ𝑥𝜓))) | ||
Theorem | bj-nexdt 33276 | Closed form of nexd 2207. (Contributed by BJ, 20-Oct-2019.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓))) | ||
Theorem | bj-nexdvt 33277* | Closed form of nexdv 1979. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) → (𝜑 → ¬ ∃𝑥𝜓)) | ||
Theorem | bj-alexbiex 33278 | Adding a second quantifier is a tranparent operation, (∀∃ case). (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | bj-exexbiex 33279 | Adding a second quantifier is a tranparent operation, (∃∃ case). (Contributed by BJ, 20-Oct-2019.) |
⊢ (∃𝑥∃𝑥𝜑 ↔ ∃𝑥𝜑) | ||
Theorem | bj-alalbial 33280 | Adding a second quantifier is a tranparent operation, (∀∀ case). (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | bj-exalbial 33281 | Adding a second quantifier is a tranparent operation, (∃∀ case). (Contributed by BJ, 20-Oct-2019.) |
⊢ (∃𝑥∀𝑥𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | bj-19.9htbi 33282 | Strengthening 19.9ht 2295 by replacing its succedent with a biconditional (19.9t 2189 does have a biconditional succedent). This propagates. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 ↔ 𝜑)) | ||
Theorem | bj-hbntbi 33283 | Strengthening hbnt 2269 by replacing its succedent with a biconditional. See also hbntg 32299 and hbntal 39717. (Contributed by BJ, 20-Oct-2019.) Proved from bj-19.9htbi 33282. (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 ↔ ∀𝑥 ¬ 𝜑)) | ||
Theorem | bj-biexal1 33284 | A general FOL biconditional that generalizes 19.9ht 2295 among others. For this and the following theorems, see also 19.35 1924, 19.21 2192, 19.23 2197. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | bj-biexal2 33285 | A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(∃𝑥𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | bj-biexal3 33286 | A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(∃𝑥𝜑 → 𝜓)) | ||
Theorem | bj-bialal 33287 | A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | bj-biexex 33288 | A general FOL biconditional. (Contributed by BJ, 20-Oct-2019.) |
⊢ (∀𝑥(𝜑 → ∃𝑥𝜓) ↔ (∃𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | bj-hbext 33289 | Closed form of hbex 2301. (Contributed by BJ, 10-Oct-2019.) |
⊢ (∀𝑦∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) | ||
Theorem | bj-nfalt 33290 | Closed form of nfal 2299. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑) | ||
Theorem | bj-nfext 33291 | Closed form of nfex 2300. (Contributed by BJ, 10-Oct-2019.) |
⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∃𝑥𝜑) | ||
Theorem | bj-eeanvw 33292* | Version of exdistrv 1998 with a disjoint variable condition on 𝑥, 𝑦 not requiring ax-11 2150. (The same can be done with eeeanv 2319 and ee4anv 2320.) (Contributed by BJ, 29-Sep-2019.) (Proof modification is discouraged.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | ||
Theorem | bj-modal4e 33293 | Dual statement of hba1 2268 (which is modal-4 ). (Contributed by BJ, 21-Dec-2020.) |
⊢ (∃𝑥∃𝑥𝜑 → ∃𝑥𝜑) | ||
Theorem | bj-modalb 33294 | A short form of the axiom B of modal logic. (Contributed by BJ, 4-Apr-2021.) |
⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | bj-axc10 33295 | Alternate (shorter) proof of axc10 2349. One can prove a version with DV(x,y) without ax-13 2334, by using ax6ev 2023 instead of ax6e 2347. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | bj-alequex 33296 | A fol lemma. See bj-alequexv 33244 for a version with a disjoint variable condition requiring fewer axioms. Can be used to reduce the proof of spimt 2350 from 133 to 112 bytes. (Contributed by BJ, 6-Oct-2018.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | ||
Theorem | bj-spimt2 33297 | A step in the proof of spimt 2350. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → 𝜓))) | ||
Theorem | bj-cbv3ta 33298 | Closed form of cbv3 2362. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦(∃𝑥𝜓 → 𝜓) ∧ ∀𝑥(𝜑 → ∀𝑦𝜑)) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-cbv3tb 33299 | Closed form of cbv3 2362. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((∀𝑦Ⅎ𝑥𝜓 ∧ ∀𝑥Ⅎ𝑦𝜑) → (∀𝑥𝜑 → ∀𝑦𝜓))) | ||
Theorem | bj-hbsb3t 33300 | A theorem close to a closed form of hbsb3 2440. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) |
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