Home | Intuitionistic Logic Explorer Theorem List (p. 17 of 141) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | exancom 1601 | Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.) |
Theorem | alrimdd 1602 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | alrimd 1603 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | eximdh 1604 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.) |
Theorem | eximd 1605 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nexd 1606 | Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.) |
Theorem | exbidh 1607 | Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Theorem | albid 1608 | Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | exbid 1609 | Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | exsimpl 1610 | Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | exsimpr 1611 | Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | alexdc 1612 | Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1638. (Contributed by Jim Kingdon, 2-Jun-2018.) |
DECID | ||
Theorem | 19.29 1613 | Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
Theorem | 19.29r 1614 | Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
Theorem | 19.29r2 1615 | Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.) |
Theorem | 19.29x 1616 | Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.) |
Theorem | 19.35-1 1617 | Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic. (Contributed by Mario Carneiro, 2-Feb-2015.) |
Theorem | 19.35i 1618 | Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Theorem | 19.25 1619 | Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Theorem | 19.30dc 1620 | Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.) |
DECID | ||
Theorem | 19.43 1621 | Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |
Theorem | 19.33b2 1622 | The antecedent provides a condition implying the converse of 19.33 1477. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1623 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.) |
Theorem | 19.33bdc 1623 | Converse of 19.33 1477 given and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version which does not require a decidability condition, see 19.33b2 1622 (Contributed by Jim Kingdon, 23-Apr-2018.) |
DECID | ||
Theorem | 19.40 1624 | Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.40-2 1625 | Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
Theorem | exintrbi 1626 | Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.) |
Theorem | exintr 1627 | Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.) |
Theorem | alsyl 1628 | Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.) |
Theorem | hbex 1629 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Theorem | nfex 1630 | If is not free in , it is not free in . (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
Theorem | 19.2 1631 | Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.) |
Theorem | i19.24 1632 | Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1617, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.) |
Theorem | i19.39 1633 | Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1617, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.) |
Theorem | 19.9ht 1634 | A closed version of one direction of 19.9 1637. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.9t 1635 | A closed version of 19.9 1637. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.) |
Theorem | 19.9h 1636 | 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.) |
Theorem | 19.9 1637 | 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.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
Theorem | alexim 1638 | One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1612. (Contributed by Jim Kingdon, 2-Jul-2018.) |
Theorem | exnalim 1639 | One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.) |
Theorem | exanaliim 1640 | A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.) |
Theorem | alexnim 1641 | A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.) |
Theorem | nnal 1642 | The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.) |
Theorem | ax6blem 1643 | If is not free in , it is not free in . This theorem doesn't use ax6b 1644 compared to hbnt 1646. (Contributed by GD, 27-Jan-2018.) |
Theorem | ax6b 1644 |
Quantified Negation. Axiom C5-2 of [Monk2] p.
113.
(Contributed by GD, 27-Jan-2018.) |
Theorem | hbn1 1645 | is not free in . (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.) |
Theorem | hbnt 1646 | Closed theorem version of bound-variable hypothesis builder hbn 1647. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Theorem | hbn 1647 | If is not free in , it is not free in . (Contributed by NM, 5-Aug-1993.) |
Theorem | hbnd 1648 | Deduction form of bound-variable hypothesis builder hbn 1647. (Contributed by NM, 3-Jan-2002.) |
Theorem | nfnt 1649 | If is not free in , then it is not free in . (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) |
Theorem | nfnd 1650 | Deduction associated with nfnt 1649. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nfn 1651 | Inference associated with nfnt 1649. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Theorem | nfdc 1652 | If is not free in , it is not free in DECID . (Contributed by Jim Kingdon, 11-Mar-2018.) |
DECID | ||
Theorem | modal-5 1653 | The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.) |
Theorem | 19.9d 1654 | A deduction version of one direction of 19.9 1637. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
Theorem | 19.9hd 1655 | A deduction version of one direction of 19.9 1637. This is an older variation of this theorem; new proofs should use 19.9d 1654. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
Theorem | excomim 1656 | One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
Theorem | excom 1657 | Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.12 1658 | Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.19 1659 | Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Theorem | 19.21-2 1660 | Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.) |
Theorem | nf2 1661 | An alternate definition of df-nf 1454, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nf3 1662 | An alternate definition of df-nf 1454. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Theorem | nf4dc 1663 | Variable is effectively not free in iff is always true or always false, given a decidability condition. The reverse direction, nf4r 1664, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.) |
DECID | ||
Theorem | nf4r 1664 | If is always true or always false, then variable is effectively not free in . The converse holds given a decidability condition, as seen at nf4dc 1663. (Contributed by Jim Kingdon, 21-Jul-2018.) |
Theorem | 19.36i 1665 | Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
Theorem | 19.36-1 1666 | Closed form of 19.36i 1665. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.) |
Theorem | 19.37-1 1667 | One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.) |
Theorem | 19.37aiv 1668* | Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.38 1669 | Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.23t 1670 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
Theorem | 19.23 1671 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
Theorem | 19.32dc 1672 | Theorem 19.32 of [Margaris] p. 90, where is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.) |
DECID | ||
Theorem | 19.32r 1673 | One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if is decidable, as seen at 19.32dc 1672. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Theorem | 19.31r 1674 | One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.) |
Theorem | 19.44 1675 | Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Theorem | 19.45 1676 | Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
Theorem | 19.34 1677 | Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Theorem | 19.41h 1678 | Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1679 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Theorem | 19.41 1679 | Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.) |
Theorem | 19.42h 1680 | Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1681 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.) |
Theorem | 19.42 1681 | Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
Theorem | excom13 1682 | Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
Theorem | exrot3 1683 | Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
Theorem | exrot4 1684 | Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) |
Theorem | nexr 1685 | Inference from 19.8a 1583. (Contributed by Jeff Hankins, 26-Jul-2009.) |
Theorem | exan 1686 | Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Theorem | hbexd 1687 | Deduction form of bound-variable hypothesis builder hbex 1629. (Contributed by NM, 2-Jan-2002.) |
Theorem | eeor 1688 | Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.) |
Theorem | a9e 1689 | 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-5 1440 through ax-14 2144 and ax-17 1519, all axioms other than ax-9 1524 are believed to be theorems of free logic, although the system without ax-9 1524 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | a9ev 1690* | At least one individual exists. Weaker version of a9e 1689. (Contributed by NM, 3-Aug-2017.) |
Theorem | ax9o 1691 | An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
Theorem | spimfv 1692* | Specialization, using implicit substitution. Version of spim 1731 with a disjoint variable condition. See spimv 1804 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.) |
Theorem | chvarfv 1693* | Implicit substitution of for into a theorem. Version of chvar 1750 with a disjoint variable condition. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.) |
Theorem | equid 1694 |
Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68.
This is often an axiom of equality in textbook systems, but we don't
need it as an axiom since it can be proved from our other axioms.
This proof is similar to Tarski's and makes use of a dummy variable . It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.) |
Theorem | nfequid 1695 | Bound-variable hypothesis builder for . This theorem tells us that any variable, including , is effectively not free in , even though is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.) |
Theorem | stdpc6 1696 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1763.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.) |
Theorem | equcomi 1697 | Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) |
Theorem | ax6evr 1698* | A commuted form of a9ev 1690. The naming reflects how axioms were numbered in the Metamath Proof Explorer as of 2020 (a numbering which we eventually plan to adopt here too, but until this happens everywhere only some theorems will have it). (Contributed by BJ, 7-Dec-2020.) |
Theorem | equcom 1699 | Commutative law for equality. (Contributed by NM, 20-Aug-1993.) |
Theorem | equcomd 1700 | Deduction form of equcom 1699, symmetry of equality. For the versions for classes, see eqcom 2172 and eqcomd 2176. (Contributed by BJ, 6-Oct-2019.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |