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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dral2 1601 | 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 → (φ ↔ ψ)) ⇒ ⊢ (∀x x = y → (∀zφ ↔ ∀zψ)) | ||
Theorem | drex2 1602 | 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 → (φ ↔ ψ)) ⇒ ⊢ (∀x x = y → (∃zφ ↔ ∃zψ)) | ||
Theorem | drnf1 1603 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
⊢ (∀x x = y → (φ ↔ ψ)) ⇒ ⊢ (∀x x = y → (Ⅎxφ ↔ Ⅎyψ)) | ||
Theorem | drnf2 1604 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
⊢ (∀x x = y → (φ ↔ ψ)) ⇒ ⊢ (∀x x = y → (Ⅎzφ ↔ Ⅎzψ)) | ||
Theorem | spimth 1605 | Closed theorem form of spim 1608. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.) |
⊢ (∀x((ψ → ∀xψ) ∧ (x = y → (φ → ψ))) → (∀xφ → ψ)) | ||
Theorem | spimt 1606 | Closed theorem form of spim 1608. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.) |
⊢ ((Ⅎxψ ∧ ∀x(x = y → (φ → ψ))) → (∀xφ → ψ)) | ||
Theorem | spimh 1607 | Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1608 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.) |
⊢ (ψ → ∀xψ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (∀xφ → ψ) | ||
Theorem | spim 1608 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1608 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.) |
⊢ Ⅎxψ & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (∀xφ → ψ) | ||
Theorem | spimeh 1609 | Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.) |
⊢ (φ → ∀xφ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (φ → ∃xψ) | ||
Theorem | spimed 1610 | Deduction version of spime 1611. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) |
⊢ (χ → Ⅎxφ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (χ → (φ → ∃xψ)) | ||
Theorem | spime 1611 | Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) |
⊢ Ⅎxφ & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (φ → ∃xψ) | ||
Theorem | cbv3 1612 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
⊢ Ⅎyφ & ⊢ Ⅎxψ & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (∀xφ → ∀yψ) | ||
Theorem | cbv3h 1613 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
⊢ (φ → ∀yφ) & ⊢ (ψ → ∀xψ) & ⊢ (x = y → (φ → ψ)) ⇒ ⊢ (∀xφ → ∀yψ) | ||
Theorem | cbv1 1614 | Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
⊢ Ⅎxφ & ⊢ Ⅎyφ & ⊢ (φ → Ⅎyψ) & ⊢ (φ → Ⅎxχ) & ⊢ (φ → (x = y → (ψ → χ))) ⇒ ⊢ (φ → (∀xψ → ∀yχ)) | ||
Theorem | cbv1h 1615 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.) |
⊢ (φ → (ψ → ∀yψ)) & ⊢ (φ → (χ → ∀xχ)) & ⊢ (φ → (x = y → (ψ → χ))) ⇒ ⊢ (∀x∀yφ → (∀xψ → ∀yχ)) | ||
Theorem | cbv2h 1616 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → (ψ → ∀yψ)) & ⊢ (φ → (χ → ∀xχ)) & ⊢ (φ → (x = y → (ψ ↔ χ))) ⇒ ⊢ (∀x∀yφ → (∀xψ ↔ ∀yχ)) | ||
Theorem | cbv2 1617 | Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
⊢ Ⅎxφ & ⊢ Ⅎyφ & ⊢ (φ → Ⅎyψ) & ⊢ (φ → Ⅎxχ) & ⊢ (φ → (x = y → (ψ ↔ χ))) ⇒ ⊢ (φ → (∀xψ ↔ ∀yχ)) | ||
Theorem | cbvalh 1618 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → ∀yφ) & ⊢ (ψ → ∀xψ) & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∀xφ ↔ ∀yψ) | ||
Theorem | cbval 1619 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
⊢ Ⅎyφ & ⊢ Ⅎxψ & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∀xφ ↔ ∀yψ) | ||
Theorem | cbvexh 1620 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.) |
⊢ (φ → ∀yφ) & ⊢ (ψ → ∀xψ) & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃xφ ↔ ∃yψ) | ||
Theorem | cbvex 1621 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ Ⅎyφ & ⊢ Ⅎxψ & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ (∃xφ ↔ ∃yψ) | ||
Theorem | chvar 1622 | Implicit substitution of y for x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
⊢ Ⅎxψ & ⊢ (x = y → (φ ↔ ψ)) & ⊢ φ ⇒ ⊢ ψ | ||
Theorem | equvini 1623 | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require z to be distinct from x and y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (x = y → ∃z(x = z ∧ z = y)) | ||
Theorem | equveli 1624 | A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1623.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
⊢ (∀z(z = x ↔ z = y) → x = y) | ||
Theorem | nfald 1625 | If x is not free in φ, it is not free in ∀yφ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) |
⊢ Ⅎyφ & ⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → Ⅎx∀yψ) | ||
Theorem | nfexd 1626 | If x is not free in φ, it is not free in ∃yφ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.) |
⊢ Ⅎyφ & ⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → Ⅎx∃yψ) | ||
Syntax | wsb 1627 | Extend wff definition to include proper substitution (read "the wff that results when y is properly substituted for x in wff φ"). (Contributed by NM, 24-Jan-2006.) |
wff [y / x]φ | ||
Definition | df-sb 1628 |
Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the
preprint). For our notation, we use [y / x]φ to mean "the wff
that results when y is properly substituted for x in the wff
φ."
We can also use [y /
x]φ in place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 1640.
Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "φ(y) is the wff that results when y is properly substituted for x in φ(x)." For example, if the original φ(x) is x = y, then φ(y) is y = y, from which we obtain that φ(x) is x = x. So what exactly does φ(x) mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1703, sbcom2 1845 and sbid2v 1854). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1639 shows. We achieve this by having x free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 1849 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1852. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1749 and sb6 1748. In classical logic, another possible definition is (x = y ∧ φ) ∨ ∀x(x = y → φ) but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff φ. (Contributed by NM, 5-Aug-1993.) |
⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ))) | ||
Theorem | sbimi 1629 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
⊢ (φ → ψ) ⇒ ⊢ ([y / x]φ → [y / x]ψ) | ||
Theorem | sbbii 1630 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ ↔ ψ) ⇒ ⊢ ([y / x]φ ↔ [y / x]ψ) | ||
Theorem | sb1 1631 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ ([y / x]φ → ∃x(x = y ∧ φ)) | ||
Theorem | sb2 1632 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x(x = y → φ) → [y / x]φ) | ||
Theorem | sbequ1 1633 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (φ → [y / x]φ)) | ||
Theorem | sbequ2 1634 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → ([y / x]φ → φ)) | ||
Theorem | stdpc7 1635 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1573.) 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, y)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.) |
⊢ (x = y → ([x / y]φ → φ)) | ||
Theorem | sbequ12 1636 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (φ ↔ [y / x]φ)) | ||
Theorem | sbequ12r 1637 | 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 | sbequ12a 1638 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → ([y / x]φ ↔ [x / y]φ)) | ||
Theorem | sbid 1639 | An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ ([x / x]φ ↔ φ) | ||
Theorem | stdpc4 1640 | The specialization axiom of standard predicate calculus. It states that if a statement φ holds for all x, then it also holds for the specific case of y (properly) substituted for x. Translated to traditional notation, it can be read: "∀xφ(x) → φ(y), provided that y is free for x in φ(x)." Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀xφ → [y / x]φ) | ||
Theorem | sbh 1641 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.) |
⊢ (φ → ∀xφ) ⇒ ⊢ ([y / x]φ ↔ φ) | ||
Theorem | sbf 1642 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎxφ ⇒ ⊢ ([y / x]φ ↔ φ) | ||
Theorem | sbf2 1643 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
⊢ ([y / x]∀xφ ↔ ∀xφ) | ||
Theorem | sb6x 1644 | Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (φ → ∀xφ) ⇒ ⊢ ([y / x]φ ↔ ∀x(x = y → φ)) | ||
Theorem | nfs1f 1645 | 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 | hbs1f 1646 | If x is not free in φ, it is not free in [y / x]φ. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (φ → ∀xφ) ⇒ ⊢ ([y / x]φ → ∀x[y / x]φ) | ||
Theorem | sbequ5 1647 | Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.) |
⊢ ([w / z]∀x x = y ↔ ∀x x = y) | ||
Theorem | sbequ6 1648 | Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.) |
⊢ ([w / z] ¬ ∀x x = y ↔ ¬ ∀x x = y) | ||
Theorem | sbt 1649 | A substitution into a theorem remains true. (See chvar 1622 and chvarv 1794 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ φ ⇒ ⊢ [y / x]φ | ||
Theorem | equsb1 1650 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
⊢ [y / x]x = y | ||
Theorem | equsb2 1651 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
⊢ [y / x]y = x | ||
Theorem | sbiedh 1652 | Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1655). New proofs should use sbied 1653 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (χ → ∀xχ)) & ⊢ (φ → (x = y → (ψ ↔ χ))) ⇒ ⊢ (φ → ([y / x]ψ ↔ χ)) | ||
Theorem | sbied 1653 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1656). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎxφ & ⊢ (φ → Ⅎxχ) & ⊢ (φ → (x = y → (ψ ↔ χ))) ⇒ ⊢ (φ → ([y / x]ψ ↔ χ)) | ||
Theorem | sbiedv 1654* | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1656). (Contributed by NM, 7-Jan-2017.) |
⊢ ((φ ∧ x = y) → (ψ ↔ χ)) ⇒ ⊢ (φ → ([y / x]ψ ↔ χ)) | ||
Theorem | sbieh 1655 | Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1656 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
⊢ (ψ → ∀xψ) & ⊢ (x = y → (φ ↔ ψ)) ⇒ ⊢ ([y / x]φ ↔ ψ) | ||
Theorem | sbie 1656 | 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 | equs5a 1657 | A property related to substitution that unlike equs5 1692 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ (∃x(x = y ∧ ∀yφ) → ∀x(x = y → φ)) | ||
Theorem | equs5e 1658 | A property related to substitution that unlike equs5 1692 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 1659 | Analogue to ax-11 1378 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 1660 | Quantifier Substitution for existential quantifiers. Analogue to ax10o 1585 but for ∃ rather than ∀. (Contributed by Jim Kingdon, 21-Dec-2017.) |
⊢ (∀x x = y → (∃xψ → ∃yψ)) | ||
Theorem | drex1 1661 | 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 1662 | 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 1663 | 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 1664 | A version of sb4 1695 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ ([y / x]∀yφ → ∀x(x = y → φ)) | ||
Theorem | equs45f 1665 | 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 1666 | 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 1667 | 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 1668 | One direction of a simplified definition of substitution that unlike sb4 1695 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ ([y / x]φ → ∀x(x = y → ∃yφ)) | ||
Theorem | hbsb2a 1669 | 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 1670 | 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 1671 | 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 1672 | 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 1673 | Version of sbco 1824 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 1674* | A version of spim 1608 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 1675* | A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1677. (Contributed by NM, 8-Nov-2006.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
⊢ (∀x x = y → ∀z w = v) | ||
Theorem | ax16 1676* |
Theorem showing that ax-16 1677 is redundant if ax-17 1400 is included in the
axiom system. The important part of the proof is provided by aev 1675.
See ax16ALT 1721 for an alternate proof that does not require ax-10 1377 or ax-12 1383. This theorem should not be referenced in any proof. Instead, use ax-16 1677 below so that theorems needing ax-16 1677 can be more easily identified. (Contributed by NM, 8-Nov-2006.) |
⊢ (∀x x = y → (φ → ∀xφ)) | ||
Axiom | ax-16 1677* |
Axiom of Distinct Variables. The only axiom of predicate calculus
requiring that variables be distinct (if we consider ax-17 1400 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 1400; see theorem ax16 1676. This axiom is obsolete and should no longer be used. It is proved above as theorem ax16 1676. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀x x = y → (φ → ∀xφ)) | ||
Theorem | dveeq2 1678* | 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 1679* | Quantifier introduction when one pair of variables is distinct. Like dveeq2 1678 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 1680* | Proof of dvelimf 1873 using dveeq2 1678 (shown as the last hypothesis) instead of ax-12 1383. This shows that ax-12 1383 could be replaced by dveeq2 1678 (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 1681* | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 8-Jan-2002.) |
⊢ (¬ ∀y y = x → (z = y → ∀x z = y)) | ||
Theorem | exlimdv 1682* | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∃xψ → χ)) | ||
Theorem | ax11v2 1683* | Recovery of ax11o 1685 from ax11v 1690 without using ax-11 1378. The hypothesis is even weaker than ax11v 1690, 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 1685. (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 1684* | Derive ax-11o 1686 from a hypothesis in the form of ax-11 1378. The hypothesis is even weaker than ax-11 1378, 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 1685. (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 1685 |
Derivation of set.mm's original ax-11o 1686 from the shorter ax-11 1378 that
has replaced it.
An open problem is whether this theorem can be proved without relying on ax-16 1677 or ax-17 1400. Normally, ax11o 1685 should be used rather than ax-11o 1686, 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 1686 |
Axiom ax-11o 1686 ("o" for "old") was the
original version of ax-11 1378,
before it was discovered (in Jan. 2007) that the shorter ax-11 1378 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 1685. This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1685. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ)))) | ||
Theorem | albidv 1687* | Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∀xψ ↔ ∀xχ)) | ||
Theorem | exbidv 1688* | Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (∃xψ ↔ ∃xχ)) | ||
Theorem | ax11b 1689 | A bidirectional version of ax-11o 1686. (Contributed by NM, 30-Jun-2006.) |
⊢ ((¬ ∀x x = y ∧ x = y) → (φ ↔ ∀x(x = y → φ))) | ||
Theorem | ax11v 1690* | This is a version of ax-11o 1686 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 1691* | Analogue to ax11v 1690 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.) |
⊢ (x = y → (∃x(x = y ∧ φ) → φ)) | ||
Theorem | equs5 1692 | 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 1693 | Lemma used in proofs of substitution properties. Like equs5 1692 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 1694 | 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 1695 | 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 1696 | One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1695 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 1697 | 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 1698 | 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 1699 | Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (¬ ∀x x = y → ([y / x]φ → ∀x[y / x]φ)) | ||
Theorem | nfsb2or 1700 | Bound-variable hypothesis builder for substitution. Similar to hbsb2 1699 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) |
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