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Type | Label | Description |
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Statement | ||
Theorem | 19.9t 2201 | Closed form of 19.9 2202 and version of 19.3t 2198 with an existential quantifier. (Contributed by NM, 13-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 14-Jul-2020.) |
⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | ||
Theorem | 19.9 2202 | A wff may be existentially quantified with a variable not free in it. Version of 19.3 2199 with an existential quantifier. Theorem 19.9 of [Margaris] p. 89. See 19.9v 1980 for a version requiring fewer axioms. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) Revised to shorten other proofs. (Revised by Wolf Lammen, 14-Jul-2020.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.21t 2203 | Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2204. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1780 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
Theorem | 19.21 2204 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1936 for a version requiring fewer axioms. See also 19.21h 2285. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1780 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | ||
Theorem | stdpc5 2205 | An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎ𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑". With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 2009. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. See stdpc5v 1935 for a version requiring fewer axioms. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) Remove dependency on ax-10 2138. (Revised by Wolf Lammen, 4-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) | ||
Theorem | 19.21-2 2206 | Version of 19.21 2204 with two quantifiers. (Contributed by NM, 4-Feb-2005.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓)) | ||
Theorem | 19.23t 2207 | Closed form of Theorem 19.23 of [Margaris] p. 90. See 19.23 2208. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 13-Aug-2020.) df-nf 1780 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 8-Oct-2022.) |
⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
Theorem | 19.23 2208 | Theorem 19.23 of [Margaris] p. 90. See 19.23v 1939 for a version requiring fewer axioms. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) | ||
Theorem | alimd 2209 | Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1806. See alimdh 1813, alimdv 1913 for variants requiring fewer axioms. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) | ||
Theorem | alrimi 2210 | Inference form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2204. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜓) | ||
Theorem | alrimdd 2211 | Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2204. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | ||
Theorem | alrimd 2212 | Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2204. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | ||
Theorem | eximd 2213 | Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1830. (Contributed by NM, 29-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | exlimi 2214 | Inference associated with 19.23 2208. See exlimiv 1927 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥𝜑 → 𝜓) | ||
Theorem | exlimd 2215 | Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 23-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) | ||
Theorem | exlimimdd 2216 | Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2217. (Revised by Wolf Lammen, 3-Sep-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝜑 → ∃𝑥𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | exlimdd 2217 | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 3-Sep-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝜑 → ∃𝑥𝜓) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | nexd 2218 | Deduction for generalization rule for negated wff. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → ¬ ∃𝑥𝜓) | ||
Theorem | albid 2219 | Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) | ||
Theorem | exbid 2220 | Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) | ||
Theorem | nfbidf 2221 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) df-nf 1780 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) | ||
Theorem | 19.16 2222 | Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓)) | ||
Theorem | 19.17 2223 | Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓)) | ||
Theorem | 19.27 2224 | Theorem 19.27 of [Margaris] p. 90. See 19.27v 1986 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.28 2225 | Theorem 19.28 of [Margaris] p. 90. See 19.28v 1987 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 7-May-2025.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | 19.19 2226 | Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓)) | ||
Theorem | 19.36 2227 | Theorem 19.36 of [Margaris] p. 90. See 19.36v 1984 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓)) | ||
Theorem | 19.36i 2228 | Inference associated with 19.36 2227. See 19.36iv 1943 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.) |
⊢ Ⅎ𝑥𝜓 & ⊢ ∃𝑥(𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | 19.37 2229 | Theorem 19.37 of [Margaris] p. 90. See 19.37v 1988 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓)) | ||
Theorem | 19.32 2230 | Theorem 19.32 of [Margaris] p. 90. See 19.32v 1937 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)) | ||
Theorem | 19.31 2231 | Theorem 19.31 of [Margaris] p. 90. See 19.31v 1938 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ 𝜓)) | ||
Theorem | 19.41 2232 | Theorem 19.41 of [Margaris] p. 90. See 19.41v 1946 for a version requiring fewer axioms. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.42 2233 | Theorem 19.42 of [Margaris] p. 90. See 19.42v 1950 for a version requiring fewer axioms. See exan 1859 for an immediate version. (Contributed by NM, 18-Aug-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) | ||
Theorem | 19.44 2234 | Theorem 19.44 of [Margaris] p. 90. See 19.44v 1989 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓)) | ||
Theorem | 19.45 2235 | Theorem 19.45 of [Margaris] p. 90. See 19.45v 1990 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) | ||
Theorem | spimfv 2236* | Specialization, using implicit substitution. Version of spim 2389 with a disjoint variable condition, which does not require ax-13 2374. See spimvw 1992 for a version with two disjoint variable conditions, requiring fewer axioms, and spimv 2392 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | chvarfv 2237* | Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 2397 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | cbv3v2 2238* | Version of cbv3 2399 with two disjoint variable conditions, which does not require ax-11 2154 nor ax-13 2374. (Contributed by BJ, 24-Jun-2019.) (Proof shortened by Wolf Lammen, 30-Aug-2021.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | sbalex 2239* |
Equivalence of two ways to express proper substitution of a setvar for
another setvar disjoint from it in a formula. This proof of their
equivalence does not use df-sb 2062.
That both sides of the biconditional express proper substitution is proved by sb5 2273 and sb6 2082. The implication "to the left" is equs4v 1996 and does not require ax-10 2138 nor ax-12 2174. It also holds without disjoint variable condition if we allow more axioms (see equs4 2418). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2462 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2461 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2265 in place of equsex 2420 in order to remove dependency on ax-13 2374. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2062. (Revised by BJ, 21-Sep-2024.) (Proof shortened by SN, 14-Aug-2025.) |
⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) | ||
Theorem | sbalexOLD 2240* | Obsolete version of sbalex 2239 as of 14-Aug-2025. (Contributed by NM, 14-Apr-2008.) (Revised by BJ, 20-Dec-2020.) (Revised by BJ, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) | ||
Theorem | sb4av 2241* | Version of sb4a 2482 with a disjoint variable condition, which does not require ax-13 2374. The distinctor antecedent from sb4b 2477 is replaced by a disjoint variable condition in this theorem. (Contributed by NM, 2-Feb-2007.) (Revised by BJ, 15-Dec-2023.) |
⊢ ([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑)) | ||
Theorem | sbimd 2242 | Deduction substituting both sides of an implication. (Contributed by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2062. (Revised by Steven Nguyen, 9-Jul-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)) | ||
Theorem | sbbid 2243 | Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) Remove dependency on ax-10 2138 and ax-13 2374. (Revised by Wolf Lammen, 24-Nov-2022.) Revise df-sb 2062. (Revised by Steven Nguyen, 11-Jul-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒)) | ||
Theorem | 2sbbid 2244 | Deduction doubly substituting both sides of a biconditional. (Contributed by AV, 30-Jul-2023.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (𝜑 → ([𝑡 / 𝑥][𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜒)) | ||
Theorem | sbequ1 2245 | An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2062. (Revised by BJ, 22-Dec-2020.) |
⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑)) | ||
Theorem | sbequ2 2246 | An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2062. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.) |
⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑)) | ||
Theorem | stdpc7 2247 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 2024.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑥)". Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑)) | ||
Theorem | sbequ12 2248 | An equality theorem for substitution. (Contributed by NM, 14-May-1993.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | ||
Theorem | sbequ12r 2249 | An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑)) | ||
Theorem | sbelx 2250* | Elimination of substitution. Also see sbel2x 2476. (Contributed by NM, 5-Aug-1993.) Avoid ax-13 2374. (Revised by Wolf Lammen, 6-Aug-2023.) Avoid ax-10 2138. (Revised by GG, 20-Aug-2023.) |
⊢ (𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ [𝑥 / 𝑦]𝜑)) | ||
Theorem | sbequ12a 2251 | An equality theorem for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Jun-2019.) |
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)) | ||
Theorem | sbid 2252 | An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.) |
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | sbcov 2253* | A composition law for substitution. Version of sbco 2509 with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by GG, 7-Aug-2023.) (Proof shortened by SN, 26-Aug-2025.) |
⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbcovOLD 2254* | Obsolete version of sbcov 2253 as of 26-Aug-2025. (Contributed by NM, 14-May-1993.) (Revised by GG, 7-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sb6a 2255* | Equivalence for substitution. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 23-Sep-2018.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑥 / 𝑦]𝜑)) | ||
Theorem | sbid2vw 2256* | Reverting substitution yields the original expression. Based on fewer axioms than sbid2v 2511, at the expense of an extra distinct variable condition. (Contributed by NM, 14-May-1993.) (Revised by Wolf Lammen, 5-Aug-2023.) |
⊢ ([𝑡 / 𝑥][𝑥 / 𝑡]𝜑 ↔ 𝜑) | ||
Theorem | axc16g 2257* | Generalization of axc16 2258. Use the latter when sufficient. This proof only requires, on top of { ax-1 6-- ax-7 2004 }, Theorem ax12v 2175. (Contributed by NM, 15-May-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) Remove dependency on ax-13 2374, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 7-Jul-2021.) Shorten axc11rv 2262. (Revised by Wolf Lammen, 11-Oct-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | ||
Theorem | axc16 2258* | Proof of older axiom ax-c16 38873. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | axc16gb 2259* | Biconditional strengthening of axc16g 2257. (Contributed by NM, 15-May-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑)) | ||
Theorem | axc16nf 2260* | If dtru 5446 is false, then there is only one element in the universe, so everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.) Remove dependency on ax-11 2154. (Revised by Wolf Lammen, 9-Sep-2018.) (Proof shortened by BJ, 14-Jun-2019.) Remove dependency on ax-10 2138. (Revised by Wolf Lammen, 12-Oct-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) | ||
Theorem | axc11v 2261* | Version of axc11 2432 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2004 }, from ax12v 2175 (contrary to axc11 2432 which seems to require the full ax-12 2174 and ax-13 2374). (Contributed by NM, 16-May-2008.) (Revised by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Theorem | axc11rv 2262* | Version of axc11r 2368 with a disjoint variable condition on 𝑥 and 𝑦, which is provable, on top of { ax-1 6-- ax-7 2004 }, from ax12v 2175 (contrary to axc11 2432 which seems to require the full ax-12 2174 and ax-13 2374, and to axc11r 2368 which seems to require the full ax-12 2174). (Contributed by BJ, 6-Jul-2021.) (Proof shortened by Wolf Lammen, 11-Oct-2021.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑)) | ||
Theorem | drsb2 2263 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | ||
Theorem | equsalv 2264* | An equivalence related to implicit substitution. Version of equsal 2419 with a disjoint variable condition, which does not require ax-13 2374. See equsalvw 2000 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsexv 2265. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | equsexv 2265* | An equivalence related to implicit substitution. Version of equsex 2420 with a disjoint variable condition, which does not require ax-13 2374. See equsexvw 2001 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2264. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) Avoid ax-10 2138. (Revised by GG, 18-Nov-2024.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | equsexvOLD 2266* | Obsolete version of equsexv 2265 as of 18-Nov-2024. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | sbft 2267 | Substitution has no effect on a nonfree variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.) |
⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑)) | ||
Theorem | sbf 2268 | Substitution for a variable not free in a wff does not affect it. For a version requiring disjoint variables but fewer axioms, see sbv 2085. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | sbf2 2269 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
⊢ ([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | sbh 2270 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 14-May-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | hbs1 2271* | The setvar 𝑥 is not free in [𝑦 / 𝑥]𝜑 when 𝑥 and 𝑦 are distinct. (Contributed by NM, 26-May-1993.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | nfs1f 2272 | If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
Theorem | sb5 2273* | Alternate definition of substitution when variables are disjoint. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1v 2084 and even needs no disjoint variable condition, see sb1 2480. Theorem sb5f 2500 replaces the disjoint variable condition with a nonfreeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Revised by Wolf Lammen, 4-Sep-2023.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | equs5av 2274* | A property related to substitution that replaces the distinctor from equs5 2462 to a disjoint variable condition. Version of equs5a 2459 with a disjoint variable condition, which does not require ax-13 2374. See also sbalex 2239. (Contributed by NM, 2-Feb-2007.) (Revised by GG, 15-Dec-2023.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | 2sb5 2275* | Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) | ||
Theorem | sbco4lemOLDOLD 2276* | Obsolete version of sbco4lem 2098 as of 12-Oct-2024. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑) | ||
Theorem | dfsb7 2277* | An alternate definition of proper substitution df-sb 2062. By introducing a dummy variable 𝑦 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑡, 𝑥, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑦 effectively insulates 𝑡 from 𝑥. To achieve this, we use a chain of two substitutions in the form of sb5 2273, first 𝑦 for 𝑥 then 𝑡 for 𝑦. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2712. Theorem sb7h 2528 provides a version where 𝜑 and 𝑦 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2062. (Revised by BJ, 25-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Sep-2023.) |
⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | ||
Theorem | sbn 2278 | Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.) Revise df-sb 2062. (Revised by BJ, 25-Dec-2020.) |
⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑) | ||
Theorem | sbex 2279* | Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) |
⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) | ||
Theorem | nf5 2280 | Alternate definition of df-nf 1780. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1780 changed. (Revised by Wolf Lammen, 11-Sep-2021.) |
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | ||
Theorem | nf6 2281 | An alternate definition of df-nf 1780. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) | ||
Theorem | nf5d 2282 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Theorem | nf5di 2283 | Since the converse holds by a1i 11, this inference shows that we can represent a not-free hypothesis with either Ⅎ𝑥𝜑 (inference form) or (𝜑 → Ⅎ𝑥𝜑) (deduction form). (Contributed by NM, 17-Aug-2018.) (Proof shortened by Wolf Lammen, 10-Jul-2019.) |
⊢ (𝜑 → Ⅎ𝑥𝜑) ⇒ ⊢ Ⅎ𝑥𝜑 | ||
Theorem | 19.9h 2284 | A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) (Proof shortened by Wolf Lammen, 14-Jul-2020.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.21h 2285 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See also 19.21 2204 and 19.21v 1936. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | ||
Theorem | 19.23h 2286 | Theorem 19.23 of [Margaris] p. 90. See 19.23 2208. (Contributed by NM, 24-Jan-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) | ||
Theorem | exlimih 2287 | Inference associated with 19.23 2208. See exlimiv 1927 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥𝜑 → 𝜓) | ||
Theorem | exlimdh 2288 | Deduction form of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-Jan-1997.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜒 → ∀𝑥𝜒) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) | ||
Theorem | equsalhw 2289* | Version of equsalh 2422 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 29-Nov-2015.) (Proof shortened by Wolf Lammen, 8-Jul-2022.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | equsexhv 2290* | An equivalence related to implicit substitution. Version of equsexh 2423 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | hba1 2291 | The setvar 𝑥 is not free in ∀𝑥𝜑. This corresponds to the axiom (4) of modal logic. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Oct-2021.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | hbnt 2292 | Closed theorem version of bound-variable hypothesis builder hbn 2293. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 14-Oct-2021.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | ||
Theorem | hbn 2293 | If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) | ||
Theorem | hbnd 2294 | Deduction form of bound-variable hypothesis builder hbn 2293. (Contributed by NM, 3-Jan-2002.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓)) | ||
Theorem | hbim1 2295 | A closed form of hbim 2297. (Contributed by NM, 2-Jun-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | hbimd 2296 | Deduction form of bound-variable hypothesis builder hbim 2297. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) | ||
Theorem | hbim 2297 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 24-Jan-1993.) (Proof shortened by Mel L. O'Cat, 3-Mar-2008.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | hban 2298 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | hb3an 2299 | 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 2300 | Introduction of implication into substitution. (Contributed by NM, 14-May-1993.) |
⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) |
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