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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sbie 1801 | Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbiev 1802* | Conversion of implicit substitution to explicit substitution. Version of sbie 1801 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | equsalv 1803* | An equivalence related to implicit substitution. Version of equsal 1737 with a disjoint variable condition. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | equs5a 1804 | A property related to substitution that unlike equs5 1839 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | equs5e 1805 | A property related to substitution that unlike equs5 1839 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | ||
Theorem | ax11e 1806 | Analogue to ax-11 1516 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.) |
⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑)) | ||
Theorem | ax10oe 1807 | Quantifier Substitution for existential quantifiers. Analogue to ax10o 1725 but for ∃ rather than ∀. (Contributed by Jim Kingdon, 21-Dec-2017.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓)) | ||
Theorem | drex1 1808 | 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.) (Revised by NM, 3-Feb-2015.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓)) | ||
Theorem | drsb1 1809 | 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, 5-Aug-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑)) | ||
Theorem | exdistrfor 1810 | Distribution of existential quantifiers, with a bound-variable hypothesis saying that 𝑦 is not free in 𝜑, but 𝑥 can be free in 𝜑 (and there is no distinct variable condition on 𝑥 and 𝑦). (Contributed by Jim Kingdon, 25-Feb-2018.) |
⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥Ⅎ𝑦𝜑) ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) | ||
Theorem | sb4a 1811 | A version of sb4 1842 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | equs45f 1812 | Two ways of expressing substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 25-Apr-2008.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb6f 1813 | Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb5f 1814 | Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | sb4e 1815 | One direction of a simplified definition of substitution that unlike sb4 1842 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | ||
Theorem | hbsb2a 1816 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | hbsb2e 1817 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑) | ||
Theorem | hbsb3 1818 | If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | nfs1 1819 | If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
Theorem | sbcof2 1820 | Version of sbco 1978 where 𝑥 is not free in 𝜑. (Contributed by Jim Kingdon, 28-Dec-2017.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | spimv 1821* | A version of spim 1748 with a distinct variable requirement instead of a bound-variable hypothesis. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | aev 1822* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1824. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑤 = 𝑣) | ||
Theorem | ax16 1823* |
Theorem showing that ax-16 1824 is redundant if ax-17 1536 is included in the
axiom system. The important part of the proof is provided by aev 1822.
See ax16ALT 1869 for an alternate proof that does not require ax-10 1515 or ax12 1522. This theorem should not be referenced in any proof. Instead, use ax-16 1824 below so that theorems needing ax-16 1824 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | ||
Axiom | ax-16 1824* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1536 to be a
metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p.
16 of the preprint). It apparently does not otherwise appear in the
literature but is easily proved from textbook predicate calculus by
cases. It is a somewhat bizarre axiom since the antecedent is always
false in set theory, but nonetheless it is technically necessary as you
can see from its uses.
This axiom is redundant if we include ax-17 1536; see Theorem ax16 1823. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax16 1823. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | dveeq2 1825* | Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
Theorem | dveeq2or 1826* | Quantifier introduction when one pair of variables is distinct. Like dveeq2 1825 but connecting ∀𝑥𝑥 = 𝑦 by a disjunction rather than negation and implication makes the theorem stronger in intuitionistic logic. (Contributed by Jim Kingdon, 1-Feb-2018.) |
⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦) | ||
Theorem | dvelimfALT2 1827* | Proof of dvelimf 2025 using dveeq2 1825 (shown as the last hypothesis) instead of ax12 1522. This shows that ax12 1522 could be replaced by dveeq2 1825 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑧𝜓) & ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | nd5 1828* | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | ||
Theorem | exlimdv 1829* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) | ||
Theorem | ax11v2 1830* | Recovery of ax11o 1832 from ax11v 1837 without using ax-11 1516. The hypothesis is even weaker than ax11v 1837, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1832. (Contributed by NM, 2-Feb-2007.) |
⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | ||
Theorem | ax11a2 1831* | Derive ax-11o 1833 from a hypothesis in the form of ax-11 1516. The hypothesis is even weaker than ax-11 1516, with 𝑧 both distinct from 𝑥 and not occurring in 𝜑. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1832. (Contributed by NM, 2-Feb-2007.) |
⊢ (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | ||
Theorem | ax11o 1832 |
Derivation of set.mm's original ax-11o 1833 from the shorter ax-11 1516 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1824 or ax-17 1536. Normally, ax11o 1832 should be used rather than ax-11o 1833, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | ||
Axiom | ax-11o 1833 |
Axiom ax-11o 1833 ("o" for "old") was the
original version of ax-11 1516,
before it was discovered (in Jan. 2007) that the shorter ax-11 1516 could
replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of
the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of
[Monk2] p. 105, from which it can be proved
by cases. To understand this
theorem more easily, think of "¬ ∀𝑥𝑥 = 𝑦 →..." as informally
meaning "if 𝑥 and 𝑦 are distinct variables,
then..." The
antecedent becomes false if the same variable is substituted for 𝑥 and
𝑦, ensuring the theorem is sound
whenever this is the case. In some
later theorems, we call an antecedent of the form ¬
∀𝑥𝑥 = 𝑦 a
"distinctor."
This axiom is redundant, as shown by Theorem ax11o 1832. This axiom is obsolete and should no longer be used. It is proved above as Theorem ax11o 1832. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | ||
Theorem | albidv 1834* | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) | ||
Theorem | exbidv 1835* | Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) | ||
Theorem | ax11b 1836 | A bidirectional version of ax-11o 1833. (Contributed by NM, 30-Jun-2006.) |
⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax11v 1837* | This is a version of ax-11o 1833 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.) |
⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax11ev 1838* | Analogue to ax11v 1837 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.) |
⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜑)) | ||
Theorem | equs5 1839 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | equs5or 1840 | Lemma used in proofs of substitution properties. Like equs5 1839 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.) |
⊢ (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | sb3 1841 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑)) | ||
Theorem | sb4 1842 | One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | sb4or 1843 | One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1842 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.) |
⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | sb4b 1844 | Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | sb4bor 1845 | Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.) |
⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | hbsb2 1846 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)) | ||
Theorem | nfsb2or 1847 | Bound-variable hypothesis builder for substitution. Similar to hbsb2 1846 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.) |
⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | sbequilem 1848 | Propositional logic lemma used in the sbequi 1849 proof. (Contributed by Jim Kingdon, 1-Feb-2018.) |
⊢ (𝜑 ∨ (𝜓 → (𝜒 → 𝜃))) & ⊢ (𝜏 ∨ (𝜓 → (𝜃 → 𝜂))) ⇒ ⊢ (𝜑 ∨ (𝜏 ∨ (𝜓 → (𝜒 → 𝜂)))) | ||
Theorem | sbequi 1849 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) | ||
Theorem | sbequ 1850 | An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑)) | ||
Theorem | drsb2 1851 | 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 | spsbe 1852 | A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑) | ||
Theorem | spsbim 1853 | Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | spsbbi 1854 | Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)) | ||
Theorem | sbbidh 1855 | Deduction substituting both sides of a biconditional. New proofs should use sbbid 1856 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒)) | ||
Theorem | sbbid 1856 | Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜒)) | ||
Theorem | sbequ8 1857 | Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sbft 1858 | 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 | sbid2h 1859 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) | ||
Theorem | sbid2 1860 | An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ 𝜑) | ||
Theorem | sbidm 1861 | An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.) |
⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | ||
Theorem | sb5rf 1862 | Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥]𝜑)) | ||
Theorem | sb6rf 1863 | Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑)) | ||
Theorem | sb8h 1864 | Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8eh 1865 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8 1866 | Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | sb8e 1867 | Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) |
⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑) | ||
Theorem | ax16i 1868* | Inference with ax-16 1824 as its conclusion, that does not require ax-10 1515, ax-11 1516, or ax12 1522 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.) |
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓)) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | ax16ALT 1869* | Version of ax16 1823 that does not require ax-10 1515 or ax12 1522 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | spv 1870* | Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spimev 1871* | Distinct-variable version of spime 1751. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | speiv 1872* | Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜓 ⇒ ⊢ ∃𝑥𝜑 | ||
Theorem | equvin 1873* | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) | ||
Theorem | a16g 1874* | A generalization of Axiom ax-16 1824. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | ||
Theorem | a16gb 1875* | A generalization of Axiom ax-16 1824. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ ∀𝑧𝜑)) | ||
Theorem | a16nf 1876* | If there is only one element in the universe, then everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) | ||
Theorem | 2albidv 1877* | Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 ↔ ∀𝑥∀𝑦𝜒)) | ||
Theorem | 2exbidv 1878* | Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒)) | ||
Theorem | 3exbidv 1879* | Formula-building rule for 3 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒)) | ||
Theorem | 4exbidv 1880* | Formula-building rule for 4 existential quantifiers (deduction form). (Contributed by NM, 3-Aug-1995.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤𝜓 ↔ ∃𝑥∃𝑦∃𝑧∃𝑤𝜒)) | ||
Theorem | 19.9v 1881* | Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.) |
⊢ (∃𝑥𝜑 ↔ 𝜑) | ||
Theorem | exlimdd 1882 | Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝜑 → ∃𝑥𝜓) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | 19.21v 1883* | Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as (𝜑 → ∀𝑥𝜑) in 19.21 1593 via the use of distinct variable conditions combined with ax-17 1536. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g., euf 2041 derived from df-eu 2039. The "f" stands for "not free in" which is less restrictive than "does not occur in". (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | ||
Theorem | alrimiv 1884* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜓) | ||
Theorem | alrimivv 1885* | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥∀𝑦𝜓) | ||
Theorem | alrimdv 1886* | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | ||
Theorem | nfdv 1887* | Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Theorem | 2ax17 1888* | Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.) |
⊢ (𝜑 → ∀𝑥∀𝑦𝜑) | ||
Theorem | alimdv 1889* | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) | ||
Theorem | eximdv 1890* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | 2alimdv 1891* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜒)) | ||
Theorem | 2eximdv 1892* | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 → ∃𝑥∃𝑦𝜒)) | ||
Theorem | 19.23v 1893* | Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.) |
⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) | ||
Theorem | 19.23vv 1894* | Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.) |
⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥∃𝑦𝜑 → 𝜓)) | ||
Theorem | sb56 1895* | Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1773. (Contributed by NM, 14-Apr-2008.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb6 1896* | Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb5 1897* | Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | sbnv 1898* | Version of sbn 1962 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.) |
⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) | ||
Theorem | sbanv 1899* | Version of sban 1965 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.) |
⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)) | ||
Theorem | sborv 1900* | Version of sbor 1964 where 𝑥 and 𝑦 are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.) |
⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) |
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