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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | hb3an 2301 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
| ⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
| Theorem | sbi2 2302 | Introduction of implication into substitution. (Contributed by NM, 14-May-1993.) |
| ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) | ||
| Theorem | sbim 2303 | Implication inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.) |
| ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | ||
| Theorem | sbrim 2304 | Substitution in an implication with a variable not free in the antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) Avoid ax-10 2141. (Revised by GG, 20-Nov-2024.) |
| ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | ||
| Theorem | sbrimOLD 2305 | Obsolete version of sbrim 2304 as of 20-Nov-2024. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | ||
| Theorem | sblim 2306 | Substitution in an implication with a variable not free in the consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.) |
| ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)) | ||
| Theorem | sbor 2307 | Disjunction inside and outside of a substitution are equivalent. (Contributed by NM, 29-Sep-2002.) |
| ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) | ||
| Theorem | sbbi 2308 | Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 14-May-1993.) |
| ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) | ||
| Theorem | sblbis 2309 | Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.) |
| ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑦 / 𝑥](𝜒 ↔ 𝜑) ↔ ([𝑦 / 𝑥]𝜒 ↔ 𝜓)) | ||
| Theorem | sbrbis 2310 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) |
| ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ [𝑦 / 𝑥]𝜒)) | ||
| Theorem | sbrbif 2311 | Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
| ⊢ Ⅎ𝑥𝜒 & ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒)) | ||
| Theorem | sbnf 2312* | Move nonfree predicate in and out of substitution; see sbal 2169 and sbex 2281. (Contributed by BJ, 2-May-2019.) (Proof shortened by Wolf Lammen, 2-May-2025.) |
| ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑) | ||
| Theorem | sbnfOLD 2313* | Obsolete version of sbnf 2312 as of 2-May-2025. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑) | ||
| Theorem | sbiev 2314* | Conversion of implicit substitution to explicit substitution. Version of sbie 2507 with a disjoint variable condition, not requiring ax-13 2377. See sbievw 2093 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2141 and shorten proof. (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Jul-2025.) |
| ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
| Theorem | sbievOLD 2315* | Obsolete version of sbiev 2314 as of 24-Aug-2025. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2141 and shorten proof. (Revised by BJ, 18-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
| Theorem | sbiedw 2316* | Conversion of implicit substitution to explicit substitution (deduction version of sbiev 2314). Version of sbied 2508 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
| Theorem | axc7 2317 |
Show that the original axiom ax-c7 38886 can be derived from ax-10 2141
(hbn1 2142), sp 2183 and propositional calculus. See ax10fromc7 38896 for the
rederivation of ax-10 2141 from ax-c7 38886.
Normally, axc7 2317 should be used rather than ax-c7 38886, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.) |
| ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | ||
| Theorem | axc7e 2318 | Abbreviated version of axc7 2317 using the existential quantifier. Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Jul-2022.) |
| ⊢ (∃𝑥∀𝑥𝜑 → 𝜑) | ||
| Theorem | modal-b 2319 | The analogue in our predicate calculus of the Brouwer axiom (B) of modal logic S5. (Contributed by NM, 5-Oct-2005.) |
| ⊢ (𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑) | ||
| Theorem | 19.9ht 2320 | A closed version of 19.9h 2286. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) |
| ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | ||
| Theorem | axc4 2321 |
Show that the original axiom ax-c4 38885 can be derived from ax-4 1809
(alim 1810), ax-10 2141 (hbn1 2142), sp 2183 and propositional calculus. See
ax4fromc4 38895 for the rederivation of ax-4 1809
from ax-c4 38885.
Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.) |
| ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
| Theorem | axc4i 2322 | Inference version of axc4 2321. (Contributed by NM, 3-Jan-1993.) |
| ⊢ (∀𝑥𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) | ||
| Theorem | nfal 2323 | If 𝑥 is not free in 𝜑, then it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
| ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦𝜑 | ||
| Theorem | nfex 2324 | If 𝑥 is not free in 𝜑, then it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Reduce symbol count in nfex 2324, hbex 2325. (Revised by Wolf Lammen, 16-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃𝑦𝜑 | ||
| Theorem | hbex 2325 | If 𝑥 is not free in 𝜑, then it is not free in ∃𝑦𝜑. (Contributed by NM, 12-Mar-1993.) Reduce symbol count in nfex 2324, hbex 2325. (Revised by Wolf Lammen, 16-Oct-2021.) |
| ⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) | ||
| Theorem | nfnf 2326 | If 𝑥 is not free in 𝜑, then it is not free in Ⅎ𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
| ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥Ⅎ𝑦𝜑 | ||
| Theorem | 19.12 2327 | Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 2349 and r19.12sn 4720. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
| ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) | ||
| Theorem | nfald 2328 | Deduction form of nfal 2323. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 16-Oct-2021.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) | ||
| Theorem | nfexd 2329 | If 𝑥 is not free in 𝜓, then it is not free in ∃𝑦𝜓. (Contributed by Mario Carneiro, 24-Sep-2016.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) | ||
| Theorem | nfsbv 2330* | If 𝑧 is not free in 𝜑, then it is not free in [𝑦 / 𝑥]𝜑 when 𝑧 is disjoint from both 𝑥 and 𝑦. Version of nfsb 2528 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2377. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Oct-2024.) |
| ⊢ Ⅎ𝑧𝜑 ⇒ ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | ||
| Theorem | nfsbvOLD 2331* | Obsolete version of nfsbv 2330 as of 25-Oct-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Wolf Lammen, 7-Feb-2023.) Remove disjoint variable condition on 𝑥, 𝑦. (Revised by Steven Nguyen, 13-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑧𝜑 ⇒ ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | ||
| Theorem | sbco2v 2332* | A composition law for substitution. Version of sbco2 2516 with disjoint variable conditions but not requiring ax-13 2377. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 29-Apr-2023.) |
| ⊢ Ⅎ𝑧𝜑 ⇒ ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
| Theorem | aaan 2333 | Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2141. (Revised by GG, 21-Nov-2024.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) | ||
| Theorem | aaanOLD 2334 | Obsolete version of aaan 2333 as of 21-Nov-2024. (Contributed by NM, 12-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) | ||
| Theorem | eeor 2335 | Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2141. (Revised by GG, 21-Nov-2024.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) | ||
| Theorem | eeorOLD 2336 | Obsolete version of eeor 2335 as of 21-Nov-2024. (Contributed by NM, 8-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) | ||
| Theorem | cbv3v 2337* | Rule used to change bound variables, using implicit substitution. Version of cbv3 2402 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
| Theorem | cbv1v 2338* | Rule used to change bound variables, using implicit substitution. Version of cbv1 2407 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) | ||
| Theorem | cbv2w 2339* | Rule used to change bound variables, using implicit substitution. Version of cbv2 2408 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
| Theorem | cbvaldw 2340* | Deduction used to change bound variables, using implicit substitution. Version of cbvald 2412 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
| Theorem | cbvexdw 2341* | Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2413 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2377. (Revised by GG, 10-Jan-2024.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
| Theorem | cbv3hv 2342* | Rule used to change bound variables, using implicit substitution. Version of cbv3h 2409 with a disjoint variable condition on 𝑥, 𝑦, which does not require ax-13 2377. Was used in a proof of axc11n 2431 (but of independent interest). (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (Proof shortened by BJ, 30-Nov-2020.) |
| ⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
| Theorem | cbvalv1 2343* | Rule used to change bound variables, using implicit substitution. Version of cbval 2403 with a disjoint variable condition, which does not require ax-13 2377. See cbvalvw 2035 for a version with two more disjoint variable conditions, requiring fewer axioms, and cbvalv 2405 for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
| Theorem | cbvexv1 2344* | Rule used to change bound variables, using implicit substitution. Version of cbvex 2404 with a disjoint variable condition, which does not require ax-13 2377. See cbvexvw 2036 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2406 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
| Theorem | cbval2v 2345* | Rule used to change bound variables, using implicit substitution. Version of cbval2 2416 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 22-Dec-2003.) (Revised by BJ, 16-Jun-2019.) (Proof shortened by GG, 10-Jan-2024.) |
| ⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) | ||
| Theorem | cbvex2v 2346* | Rule used to change bound variables, using implicit substitution. Version of cbvex2 2417 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 14-Sep-2003.) (Revised by BJ, 16-Jun-2019.) |
| ⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓) | ||
| Theorem | dvelimhw 2347* | Proof of dvelimh 2455 without using ax-13 2377 but with additional distinct variable conditions. (Contributed by NM, 1-Oct-2002.) (Revised by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 23-Dec-2018.) |
| ⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑧𝜓) & ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) | ||
| Theorem | pm11.53 2348* | Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1944 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.) |
| ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) | ||
| Theorem | 19.12vv 2349* | Special case of 19.12 2327 where its converse holds. See 19.12vvv 1988 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.) |
| ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓)) | ||
| Theorem | eean 2350 | Distribute existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | ||
| Theorem | eeanv 2351* | Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1949 and 19.42v 1953. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1955. (Contributed by NM, 26-Jul-1995.) |
| ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | ||
| Theorem | eeeanv 2352* | Distribute three existential quantifiers over a conjunction. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.) |
| ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒)) | ||
| Theorem | ee4anv 2353* | Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv 1956. (Contributed by NM, 31-Jul-1995.) Remove disjoint variable conditions on 𝑦, 𝑧 and 𝑥, 𝑤. (Revised by Eric Schmidt, 26-Oct-2025.) |
| ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) | ||
| Theorem | ee4anvOLD 2354* | Obsolete version of ee4anv 2353 as of 26-Oct-2025. (Contributed by NM, 31-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) | ||
| Theorem | sb8v 2355* | Substitution of variable in universal quantifier. Version of sb8f 2356 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2177. (Contributed by SN, 5-Dec-2024.) |
| ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
| Theorem | sb8f 2356* | Substitution of variable in universal quantifier. Version of sb8 2522 with a disjoint variable condition, not requiring ax-10 2141 or ax-13 2377. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2141. (Revised by SN, 5-Dec-2024.) |
| ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
| Theorem | sb8fOLD 2357* | Obsolete version of sb8f 2356 as of 5-Dec-2024. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
| Theorem | sb8ef 2358* | Substitution of variable in existential quantifier. Version of sb8e 2523 with a disjoint variable condition, not requiring ax-13 2377. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) |
| ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | ||
| Theorem | 2sb8ef 2359* | An equivalent expression for double existence. Version of 2sb8e 2535 with more disjoint variable conditions, not requiring ax-13 2377. (Contributed by Wolf Lammen, 28-Jan-2023.) |
| ⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | ||
| Theorem | sb6rfv 2360* | Reversed substitution. Version of sb6rf 2473 requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993.) (Revised by Wolf Lammen, 7-Feb-2023.) |
| ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) | ||
| Theorem | sbnf2 2361* | Two ways of expressing "𝑥 is (effectively) not free in 𝜑". (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.) Avoid ax-13 2377. (Revised by Wolf Lammen, 30-Jan-2023.) |
| ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) | ||
| Theorem | exsb 2362* | An equivalent expression for existence. One direction (exsbim 2001) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2377. (Revised by Wolf Lammen, 16-Oct-2022.) |
| ⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
| Theorem | 2exsb 2363* | An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.) |
| ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑)) | ||
| Theorem | sbbib 2364* | Reversal of substitution. (Contributed by AV, 6-Aug-2023.) (Proof shortened by Wolf Lammen, 4-Sep-2023.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) | ||
| Theorem | sbbibvv 2365* | Reversal of substitution. (Contributed by AV, 6-Aug-2023.) |
| ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) | ||
| Theorem | cbvsbvf 2366* | Change the bound variable (i.e. the substituted one) in wff's linked by implicit substitution. The proof was part of a former cbvabw 2813 version. (Contributed by GG and WL, 26-Oct-2024.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) | ||
| Theorem | cleljustALT 2367* | Alternate proof of cleljust 2117. It is kept here and should not be modified because it is referenced on the Metamath Proof Explorer Home Page (mmset.html) as an example of how disjoint variable conditions are inherited by substitutions. (Contributed by NM, 28-Jan-2004.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) | ||
| Theorem | cleljustALT2 2368* | Alternate proof of cleljust 2117. Compared with cleljustALT 2367, it uses nfv 1914 followed by equsexv 2268 instead of ax-5 1910 followed by equsexhv 2292, so it uses the idiom Ⅎ𝑥𝜑 instead of 𝜑 → ∀𝑥𝜑 to express nonfreeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) | ||
| Theorem | equs5aALT 2369 | Alternate proof of equs5a 2462. Uses ax-12 2177 but not ax-13 2377. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
| Theorem | equs5eALT 2370 | Alternate proof of equs5e 2463. Uses ax-12 2177 but not ax-13 2377. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | ||
| Theorem | axc11r 2371 |
Same as axc11 2435 but with reversed antecedent. Note the use
of ax-12 2177
(and not merely ax12v 2178 as in axc11rv 2265).
This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2053 and aecom 2432, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations. As of 2024, it is used in conjunction with ax-13 2377 only, and like that, it should be applied only in niches where indispensable. (Contributed by NM, 25-Jul-2015.) |
| ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
| Theorem | dral1v 2372* | Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2444 with a disjoint variable condition, which does not require ax-13 2377. Remark: the corresponding versions for dral2 2443 and drex2 2447 are instances of albidv 1920 and exbidv 1921 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2178. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2141. (Revised by GG, 18-Nov-2024.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
| Theorem | dral1vOLD 2373* | Obsolete version of dral1v 2372 as of 18-Nov-2024. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2178. (Revised by Wolf Lammen, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
| Theorem | drex1v 2374* | Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drex1 2446 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 27-Feb-2005.) (Revised by BJ, 17-Jun-2019.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓)) | ||
| Theorem | drnf1v 2375* | Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2448 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2141. (Revised by GG, 18-Nov-2024.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓)) | ||
| Theorem | drnf1vOLD 2376* | Obsolete version of drnf1v 2375 as of 18-Nov-2024. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓)) | ||
| Axiom | ax-13 2377 |
Axiom of Quantified Equality. One of the equality and substitution axioms
of predicate calculus with equality.
An equivalent way to express this axiom that may be easier to understand is (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧))) (see ax13b 2031). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent ¬ 𝑥 = 𝑦 to hold, 𝑥 and 𝑦 must have different values and thus cannot be the same object-language variable (so they are effectively "distinct variables" even though no $d is present). Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-5 1910, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows. Note that ax-5 1910 cannot prove this because its distinct variable ($d) requirement is not satisfied directly but only indirectly (outside of Metamath) by the argument above. The original version of this axiom was ax-c9 38891 and was replaced with this shorter ax-13 2377 in December 2015. The old axiom is proved from this one as Theorem axc9 2387. The primary purpose of this axiom is to provide a way to introduce the quantifier ∀𝑥 on 𝑦 = 𝑧 even when 𝑥 and 𝑦 are substituted with the same variable. In this case, the first antecedent becomes ¬ 𝑥 = 𝑥 and the axiom still holds. This axiom is mostly used to eliminate conditions requiring set variables be distinct (cf. ax6ev 1969 and ax6e 2388, for example) in proofs. In practice, theorems beyond elementary set theory do not really benefit from such eliminations, so direct or indirect application of this axiom is discouraged now. You need to explicitly confirm its use in case you see a sensible application in a niche. After some assisting contributions by others over the years, it was in particular the extensive work of Gino Giotto in 2024 that helped reducing dependencies on this axiom on a large scale. Although this version is shorter, the original version axc9 2387 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2387 is in dvelimh 2455 which converts a distinct variable pair to the distinctor antecedent ¬ ∀𝑥𝑥 = 𝑦. In particular, it is conjectured that it is not possible to prove ax6 2389 from ax6v 1968 without this axiom. This axiom can be weakened if desired by adding distinct variable restrictions on pairs 𝑥, 𝑧 and 𝑦, 𝑧. To show that, we add these restrictions to Theorem ax13v 2378 and use only ax13v 2378 for further derivations. Thus, ax13v 2378 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2378 (preferred) or ax13 2380 (if the stronger form is needed). This axiom scheme is logically redundant (see ax13w 2136) but is used as an auxiliary axiom scheme to achieve scheme completeness (i.e. so that all possible cases of bundling can be proved; see text linked at mmtheorems.html#ax6dgen 2136). It is not known whether this axiom can be derived from the others. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.) |
| ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
| Theorem | ax13v 2378* |
A weaker version of ax-13 2377 with distinct variable restrictions on pairs
𝑥,
𝑧 and 𝑦, 𝑧. In order to show (with
ax13 2380) that this
weakening is still adequate, this should be the only theorem referencing
ax-13 2377 directly.
Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1910. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. Preferably, use the version ax13w 2136 to avoid the propagation of ax-13 2377. (Contributed by NM, 30-Jun-2016.) (New usage is discouraged.) |
| ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
| Theorem | ax13lem1 2379* | A version of ax13v 2378 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2380 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
| ⊢ (¬ 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
| Theorem | ax13 2380 | Derive ax-13 2377 from ax13v 2378 and Tarski's FOL. This shows that the weakening in ax13v 2378 is still sufficient for a complete system. Preferably, use the weaker ax13w 2136 to avoid the propagation of ax-13 2377. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 2-Jun-2021.) (New usage is discouraged.) |
| ⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
| Theorem | ax13lem2 2381* | Lemma for nfeqf2 2382. This lemma is equivalent to ax13v 2378 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 8-Sep-2018.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2020.) (New usage is discouraged.) |
| ⊢ (¬ 𝑥 = 𝑦 → (∃𝑥 𝑧 = 𝑦 → 𝑧 = 𝑦)) | ||
| Theorem | nfeqf2 2382* | An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2177. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.) |
| ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | ||
| Theorem | dveeq2 2383* | Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2157. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
| ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
| Theorem | nfeqf1 2384* | An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.) |
| ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) | ||
| Theorem | dveeq1 2385* | Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2157. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
| ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
| Theorem | nfeqf 2386 | A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 38891. Usage of this theorem is discouraged because it depends on ax-13 2377. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2157. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.) |
| ⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦) | ||
| Theorem | axc9 2387 | Derive set.mm's original ax-c9 38891 from the shorter ax-13 2377. Usage is discouraged to avoid uninformed ax-13 2377 propagation. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.) (New usage is discouraged.) |
| ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
| Theorem | ax6e 2388 |
At least one individual exists. This is not a theorem of free logic,
which is sound in empty domains. For such a logic, we would add this
theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the
system consisting of ax-4 1809 through ax-9 2118,
all axioms other than
ax-6 1967 are believed to be theorems of free logic,
although the system
without ax-6 1967 is not complete in free logic.
Usage of this theorem is discouraged because it depends on ax-13 2377. It is preferred to use ax6ev 1969 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2379 became available. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
| ⊢ ∃𝑥 𝑥 = 𝑦 | ||
| Theorem | ax6 2389 |
Theorem showing that ax-6 1967 follows from the weaker version ax6v 1968.
(Even though this theorem depends on ax-6 1967,
all references of ax-6 1967 are
made via ax6v 1968. An earlier version stated ax6v 1968
as a separate axiom,
but having two axioms caused some confusion.)
This theorem should be referenced in place of ax-6 1967 so that all proofs can be traced back to ax6v 1968. When possible, use the weaker ax6v 1968 rather than ax6 2389 since the ax6v 1968 derivation is much shorter and requires fewer axioms. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use ax6v 1968 instead. (New usage is discouraged.) |
| ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
| Theorem | axc10 2390 |
Show that the original axiom ax-c10 38887 can be derived from ax6 2389
and axc7 2317
(on top of propositional calculus, ax-gen 1795, and ax-4 1809). See
ax6fromc10 38897 for the rederivation of ax6 2389
from ax-c10 38887.
Normally, axc10 2390 should be used rather than ax-c10 38887, except by theorems specifically studying the latter's properties. See bj-axc10v 36794 for a weaker version requiring fewer axioms. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) Usage of this theorem is discouraged because it depends on ax-13 2377. (New usage is discouraged.) |
| ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
| Theorem | spimt 2391 | Closed theorem form of spim 2392. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Mar-2023.) Usage of this theorem is discouraged because it depends on ax-13 2377. (New usage is discouraged.) |
| ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) | ||
| Theorem | spim 2392 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2392 series of theorems requires that only one direction of the substitution hypothesis hold. Usage of this theorem is discouraged because it depends on ax-13 2377. See spimw 1970 for a version requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
| Theorem | spimed 2393 | Deduction version of spime 2394. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use spimedv 2197 instead. (New usage is discouraged.) |
| ⊢ (𝜒 → Ⅎ𝑥𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) | ||
| Theorem | spime 2394 | Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 1971 and spimevw 1994 for weaker versions requiring fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use spimefv 2198 instead. (New usage is discouraged.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
| Theorem | spimv 2395* | A version of spim 2392 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2239 and spimvw 1995 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use spimvw 1995 instead. (New usage is discouraged.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
| Theorem | spimvALT 2396* | Alternate proof of spimv 2395. Note that it requires only ax-1 6 through ax-5 1910 together with ax6e 2388. Currently, proofs derive from ax6v 1968, but if ax-6 1967 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2141. (Revised by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
| Theorem | spimev 2397* | Distinct-variable version of spime 2394. (Contributed by NM, 10-Jan-1993.) Usage of this theorem is discouraged because it depends on ax-13 2377. Use spimevw 1994 instead. (New usage is discouraged.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
| Theorem | spv 2398* | Specialization, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker spvv 1996 if possible. (Contributed by NM, 30-Aug-1993.) (New usage is discouraged.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
| Theorem | spei 2399 | Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker speiv 1972 if possible. (Contributed by NM, 19-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.) |
| ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ ∃𝑥𝜑 | ||
| Theorem | chvar 2400 | Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker chvarfv 2240 if possible. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) |
| ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
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