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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sbrim 2301 | 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 2137. (Revised by Gino Giotto, 20-Nov-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | sbrimOLD 2302 | Obsolete version of sbrim 2301 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 2303 | 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 2304 | 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 2305 | 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 2306 | 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 2307 | 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 2308 | 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 | sbiev 2309* | Conversion of implicit substitution to explicit substitution. Version of sbie 2505 with a disjoint variable condition, not requiring ax-13 2371. See sbievw 2095 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 2137 and shorten proof. (Revised by BJ, 18-Jul-2023.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbiedw 2310* | Conversion of implicit substitution to explicit substitution (deduction version of sbiev 2309). Version of sbied 2506 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | axc7 2311 |
Show that the original axiom ax-c7 37242 can be derived from ax-10 2137
(hbn1 2138) , sp 2176 and propositional calculus. See ax10fromc7 37252 for the
rederivation of ax-10 2137 from ax-c7 37242.
Normally, axc7 2311 should be used rather than ax-c7 37242, except by theorems specifically studying the latter's properties. (Contributed by NM, 21-May-2008.) |
⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | ||
Theorem | axc7e 2312 | Abbreviated version of axc7 2311 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 2313 | 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 2314 | A closed version of 19.9h 2283. (Contributed by NM, 13-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | ||
Theorem | axc4 2315 |
Show that the original axiom ax-c4 37241 can be derived from ax-4 1811
(alim 1812), ax-10 2137 (hbn1 2138), sp 2176 and propositional calculus. See
ax4fromc4 37251 for the rederivation of ax-4 1811
from ax-c4 37241.
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 2316 | Inference version of axc4 2315. (Contributed by NM, 3-Jan-1993.) |
⊢ (∀𝑥𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | nfal 2317 | 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 2318 | 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 2318, hbex 2319. (Revised by Wolf Lammen, 16-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃𝑦𝜑 | ||
Theorem | hbex 2319 | 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 2318, hbex 2319. (Revised by Wolf Lammen, 16-Oct-2021.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) | ||
Theorem | nfnf 2320 | 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 2321 | 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 2344 and r19.12sn 4679. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) | ||
Theorem | nfald 2322 | Deduction form of nfal 2317. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 16-Oct-2021.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) | ||
Theorem | nfexd 2323 | 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 2324* | 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 2526 with an additional disjoint variable condition on 𝑥, 𝑧 but not requiring ax-13 2371. (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 2325* | Obsolete version of nfsbv 2324 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 | hbsbwOLD 2326* | Obsolete version of hbsbw 2169 as of 23-May-2024. (Contributed by NM, 12-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑧𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) | ||
Theorem | sbco2v 2327* | A composition law for substitution. Version of sbco2 2514 with disjoint variable conditions but not requiring ax-13 2371. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 29-Apr-2023.) |
⊢ Ⅎ𝑧𝜑 ⇒ ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | aaan 2328 | Distribute universal quantifiers. (Contributed by NM, 12-Aug-1993.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) | ||
Theorem | aaanOLD 2329 | Obsolete version of aaan 2328 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 2330 | Distribute existential quantifiers. (Contributed by NM, 8-Aug-1994.) Avoid ax-10 2137. (Revised by Gino Giotto, 21-Nov-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) | ||
Theorem | eeorOLD 2331 | Obsolete version of eeor 2330 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 2332* | Rule used to change bound variables, using implicit substitution. Version of cbv3 2396 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | cbv1v 2333* | Rule used to change bound variables, using implicit substitution. Version of cbv1 2401 with a disjoint variable condition, which does not require ax-13 2371. (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 2334* | Rule used to change bound variables, using implicit substitution. Version of cbv2 2402 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | cbvaldw 2335* | Deduction used to change bound variables, using implicit substitution. Version of cbvald 2406 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | cbvexdw 2336* | Deduction used to change bound variables, using implicit substitution. Version of cbvexd 2407 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 2-Jan-2002.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒)) | ||
Theorem | cbv3hv 2337* | Rule used to change bound variables, using implicit substitution. Version of cbv3h 2403 with a disjoint variable condition on 𝑥, 𝑦, which does not require ax-13 2371. Was used in a proof of axc11n 2425 (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 2338* | Rule used to change bound variables, using implicit substitution. Version of cbval 2397 with a disjoint variable condition, which does not require ax-13 2371. See cbvalvw 2039 for a version with two more disjoint variable conditions, requiring fewer axioms, and cbvalv 2399 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 2339* | Rule used to change bound variables, using implicit substitution. Version of cbvex 2398 with a disjoint variable condition, which does not require ax-13 2371. See cbvexvw 2040 for a version with two disjoint variable conditions, requiring fewer axioms, and cbvexv 2400 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 2340* | Rule used to change bound variables, using implicit substitution. Version of cbval2 2410 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 22-Dec-2003.) (Revised by BJ, 16-Jun-2019.) (Proof shortened by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑧𝜑 & ⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) | ||
Theorem | cbvex2v 2341* | Rule used to change bound variables, using implicit substitution. Version of cbvex2 2411 with a disjoint variable condition, which does not require ax-13 2371. (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 2342* | Proof of dvelimh 2449 without using ax-13 2371 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 2343* | Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1947 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) | ||
Theorem | 19.12vv 2344* | Special case of 19.12 2321 where its converse holds. See 19.12vvv 1992 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 2345 | 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 2346* | Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1953 and 19.42v 1957. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1959. (Contributed by NM, 26-Jul-1995.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓)) | ||
Theorem | eeeanv 2347* | 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 2348* | Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv 1960. (Contributed by NM, 31-Jul-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤(𝜑 ∧ 𝜓) ↔ (∃𝑥∃𝑦𝜑 ∧ ∃𝑧∃𝑤𝜓)) | ||
Theorem | sb8v 2349* | Substitution of variable in universal quantifier. Version of sb8f 2350 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2171. (Contributed by SN, 5-Dec-2024.) |
⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8f 2350* | Substitution of variable in universal quantifier. Version of sb8 2520 with a disjoint variable condition, not requiring ax-10 2137 or ax-13 2371. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2137. (Revised by SN, 5-Dec-2024.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8fOLD 2351* | Obsolete version of sb8f 2350 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 2352* | Substitution of variable in existential quantifier. Version of sb8e 2521 with a disjoint variable condition, not requiring ax-13 2371. (Contributed by NM, 12-Aug-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | 2sb8ef 2353* | An equivalent expression for double existence. Version of 2sb8e 2534 with more disjoint variable conditions, not requiring ax-13 2371. (Contributed by Wolf Lammen, 28-Jan-2023.) |
⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎ𝑧𝜑 ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | sb6rfv 2354* | Reversed substitution. Version of sb6rf 2467 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 2355* | 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 2371. (Revised by Wolf Lammen, 30-Jan-2023.) |
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑦∀𝑧([𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑)) | ||
Theorem | exsb 2356* | An equivalent expression for existence. One direction (exsbim 2005) needs fewer axioms. (Contributed by NM, 2-Feb-2005.) Avoid ax-13 2371. (Revised by Wolf Lammen, 16-Oct-2022.) |
⊢ (∃𝑥𝜑 ↔ ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | 2exsb 2357* | 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 2358* | 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 2359* | Reversal of substitution. (Contributed by AV, 6-Aug-2023.) |
⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓)) | ||
Theorem | sbievg 2360* | Substitution applied to expressions linked by implicit substitution. The proof was part of a former cbvabw 2811 version. (Contributed by GG and WL, 26-Oct-2024.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) | ||
Theorem | cleljustALT 2361* | Alternate proof of cleljust 2115. 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 2362* | Alternate proof of cleljust 2115. Compared with cleljustALT 2361, it uses nfv 1917 followed by equsexv 2260 instead of ax-5 1913 followed by equsexhv 2289, 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 2363 | Alternate proof of equs5a 2456. Uses ax-12 2171 but not ax-13 2371. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | equs5eALT 2364 | Alternate proof of equs5e 2457. Uses ax-12 2171 but not ax-13 2371. (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 2365 |
Same as axc11 2429 but with reversed antecedent. Note the use
of ax-12 2171
(and not merely ax12v 2172 as in axc11rv 2257).
This theorem is mostly used to eliminate conditions requiring set variables be distinct (cf. cbvaev 2056 and aecom 2426, 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 2371 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 2366* | Formula-building lemma for use with the Distinctor Reduction Theorem. Version of dral1 2438 with a disjoint variable condition, which does not require ax-13 2371. Remark: the corresponding versions for dral2 2437 and drex2 2441 are instances of albidv 1923 and exbidv 1924 respectively. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2172. (Revised by Wolf Lammen, 30-Mar-2024.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | dral1vOLD 2367* | Obsolete version of dral1v 2366 as of 18-Nov-2024. (Contributed by NM, 24-Nov-1994.) (Revised by BJ, 17-Jun-2019.) Base the proof on ax12v 2172. (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 2368* | Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drex1 2440 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 27-Feb-2005.) (Revised by BJ, 17-Jun-2019.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓)) | ||
Theorem | drnf1v 2369* | Formula-building lemma for use with the Distinctor Reduction Theorem. Version of drnf1 2442 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 4-Oct-2016.) (Revised by BJ, 17-Jun-2019.) Avoid ax-10 2137. (Revised by Gino Giotto, 18-Nov-2024.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓)) | ||
Theorem | drnf1vOLD 2370* | Obsolete version of drnf1v 2369 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 2371 |
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 2035). 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 $d is present). Similarly, 𝑥 and 𝑧 cannot be the same object-language variable. Therefore, 𝑥 will not occur in the wff 𝑦 = 𝑧 when the first two antecedents hold, so analogous to ax-5 1913, the conclusion (𝑦 = 𝑧 → ∀𝑥𝑦 = 𝑧) follows. Note that ax-5 1913 cannot prove this because its distinct variable ($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 37247 and was replaced with this shorter ax-13 2371 in December 2015. The old axiom is proved from this one as Theorem axc9 2381. 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 1973 and ax6e 2382, 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 2381 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of axc9 2381 is in dvelimh 2449 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 2383 from ax6v 1972 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 2372 and use only ax13v 2372 for further derivations. Thus, ax13v 2372 should be the only theorem referencing this axiom. Other theorems can reference either ax13v 2372 (preferred) or ax13 2374 (if the stronger form is needed). This axiom scheme is logically redundant (see ax13w 2132) 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 2132). 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 2372* |
A weaker version of ax-13 2371 with distinct variable restrictions on pairs
𝑥,
𝑧 and 𝑦, 𝑧. In order to show (with
ax13 2374) that this
weakening is still adequate, this should be the only theorem referencing
ax-13 2371 directly.
Had we additionally required 𝑥 and 𝑦 be distinct, too, this theorem would have been a direct consequence of ax-5 1913. So essentially this theorem states, that a distinct variable condition can be replaced with an inequality between set variables. Preferably, use the version ax13w 2132 to avoid the propagation of ax-13 2371. (Contributed by NM, 30-Jun-2016.) (New usage is discouraged.) |
⊢ (¬ 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
Theorem | ax13lem1 2373* | A version of ax13v 2372 with one distinct variable restriction dropped. For convenience, 𝑦 is kept on the right side of equations. The proof of ax13 2374 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 2374 | Derive ax-13 2371 from ax13v 2372 and Tarski's FOL. This shows that the weakening in ax13v 2372 is still sufficient for a complete system. Preferably, use the weaker ax13w 2132 to avoid the propagation of ax-13 2371. (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 2375* | Lemma for nfeqf2 2376. This lemma is equivalent to ax13v 2372 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 2376* | An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Wolf Lammen, 9-Jun-2019.) Remove dependency on ax-12 2171. (Revised by Wolf Lammen, 16-Dec-2022.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦) | ||
Theorem | dveeq2 2377* | Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.) Remove dependency on ax-11 2154. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
Theorem | nfeqf1 2378* | An equation between setvar is free of any other setvar. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 = 𝑧) | ||
Theorem | dveeq1 2379* | Quantifier introduction when one pair of variables is distinct. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 2-Jan-2002.) Remove dependency on ax-11 2154. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) | ||
Theorem | nfeqf 2380 | A variable is effectively not free in an equality if it is not either of the involved variables. Ⅎ version of ax-c9 37247. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Mario Carneiro, 6-Oct-2016.) Remove dependency on ax-11 2154. (Revised by Wolf Lammen, 6-Sep-2018.) (New usage is discouraged.) |
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → Ⅎ𝑧 𝑥 = 𝑦) | ||
Theorem | axc9 2381 | Derive set.mm's original ax-c9 37247 from the shorter ax-13 2371. Usage is discouraged to avoid uninformed ax-13 2371 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 2382 |
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 1811 through ax-9 2116,
all axioms other than
ax-6 1971 are believed to be theorems of free logic,
although the system
without ax-6 1971 is not complete in free logic.
Usage of this theorem is discouraged because it depends on ax-13 2371. It is preferred to use ax6ev 1973 when it is sufficient. (Contributed by NM, 14-May-1993.) Shortened after ax13lem1 2373 became available. (Revised by Wolf Lammen, 8-Sep-2018.) (New usage is discouraged.) |
⊢ ∃𝑥 𝑥 = 𝑦 | ||
Theorem | ax6 2383 |
Theorem showing that ax-6 1971 follows from the weaker version ax6v 1972.
(Even though this theorem depends on ax-6 1971,
all references of ax-6 1971 are
made via ax6v 1972. An earlier version stated ax6v 1972
as a separate axiom,
but having two axioms caused some confusion.)
This theorem should be referenced in place of ax-6 1971 so that all proofs can be traced back to ax6v 1972. When possible, use the weaker ax6v 1972 rather than ax6 2383 since the ax6v 1972 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 2371. Use ax6v 1972 instead. (New usage is discouraged.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | axc10 2384 |
Show that the original axiom ax-c10 37243 can be derived from ax6 2383
and axc7 2311
(on top of propositional calculus, ax-gen 1797, and ax-4 1811). See
ax6fromc10 37253 for the rederivation of ax6 2383
from ax-c10 37243.
Normally, axc10 2384 should be used rather than ax-c10 37243, except by theorems specifically studying the latter's properties. See bj-axc10v 35153 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 2371. (New usage is discouraged.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | spimt 2385 | Closed theorem form of spim 2386. (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 2371. (New usage is discouraged.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spim 2386 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2386 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 2371. See spimw 1974 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 2387 | Deduction version of spime 2388. (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 2371. Use spimedv 2190 instead. (New usage is discouraged.) |
⊢ (𝜒 → Ⅎ𝑥𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) | ||
Theorem | spime 2388 | Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 1975 and spimevw 1998 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 2371. Use spimefv 2191 instead. (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | spimv 2389* | A version of spim 2386 with a distinct variable requirement instead of a bound-variable hypothesis. See spimfv 2232 and spimvw 1999 for versions requiring fewer axioms. (Contributed by NM, 31-Jul-1993.) Usage of this theorem is discouraged because it depends on ax-13 2371. Use spimvw 1999 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spimvALT 2390* | Alternate proof of spimv 2389. Note that it requires only ax-1 6 through ax-5 1913 together with ax6e 2382. Currently, proofs derive from ax6v 1972, but if ax-6 1971 could be used instead, this proof would reduce axiom usage. (Contributed by NM, 31-Jul-1993.) Remove dependency on ax-10 2137. (Revised by BJ, 29-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spimev 2391* | Distinct-variable version of spime 2388. (Contributed by NM, 10-Jan-1993.) Usage of this theorem is discouraged because it depends on ax-13 2371. Use spimevw 1998 instead. (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | spv 2392* | Specialization, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker spvv 2000 if possible. (Contributed by NM, 30-Aug-1993.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spei 2393 | Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker speiv 1976 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 2394 | Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker chvarfv 2233 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 | ||
Theorem | chvarv 2395* | Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker chvarvv 2002 if possible. (Contributed by NM, 20-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | cbv3 2396 | Rule used to change bound variables, using implicit substitution, that does not use ax-c9 37247. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbv3v 2332 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | cbval 2397 | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out cbvalw 2038, cbvalvw 2039, cbvalv1 2338 for versions requiring fewer axioms. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbvex 2398 | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. Check out cbvexvw 2040, cbvexv1 2339 for weaker versions requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Theorem | cbvalv 2399* | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. See cbvalvw 2039 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2137 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbvexv 2400* | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. See cbvexvw 2040 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2137 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
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