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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfexdc 1501 |
Defining ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | exalim 1502 | One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1501. (Contributed by Jim Kingdon, 29-Jul-2018.) |
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The equality predicate was introduced above in wceq 1353 for use by df-tru 1356. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Theorem | weq 1503 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1503 as an axiomatic statement, as was done in an
older version of this database, we introduce it by "proving" a
special
case of set theory's more general wceq 1353. This lets us avoid overloading
the |
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Axiom | ax-8 1504 |
Axiom of Equality. One of the equality and substitution axioms of
predicate calculus with equality. This is similar to, but not quite, a
transitive law for equality (proved later as equtr 1709). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1504 through ax-16 1814 are the axioms having to do with equality,
substitution, and logical properties of our binary predicate |
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Axiom | ax-10 1505 |
Axiom of Quantifier Substitution. One of the equality and substitution
axioms of predicate calculus with equality. Appears as Lemma L12 in
[Megill] p. 445 (p. 12 of the preprint).
The original version of this axiom was ax-10o 1716 ("o" for "old") and was replaced with this shorter ax-10 1505 in May 2008. The old axiom is proved from this one as Theorem ax10o 1715. Conversely, this axiom is proved from ax-10o 1716 as Theorem ax10 1717. (Contributed by NM, 5-Aug-1993.) |
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Axiom | ax-11 1506 |
Axiom of Variable Substitution. One of the 5 equality axioms of predicate
calculus. The final consequent ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1827, ax11v2 1820 and ax-11o 1823. (Contributed by NM, 5-Aug-1993.) |
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Axiom | ax-i12 1507 |
Axiom of Quantifier Introduction. One of the equality and substitution
axioms of predicate calculus with equality. Informally, it says that
whenever ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() This axiom has been modified from the original ax12 1512 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias ax12or 1508 instead, for labeling consistency. (New usage is discouraged.) |
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Theorem | ax12or 1508 | Alias for ax-i12 1507, to be used in place of it for labeling consistency. (Contributed by NM, 3-Feb-2015.) |
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Axiom | ax-bndl 1509 |
Axiom of bundling. The general idea of this axiom is that two variables
are either distinct or non-distinct. That idea could be expressed as
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability). This axiom implies ax-i12 1507 as can be seen at axi12 1514. Whether ax-bndl 1509 can be proved from the remaining axioms including ax-i12 1507 is not known. The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.) |
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Axiom | ax-4 1510 |
Axiom of Specialization. A quantified wff implies the wff without a
quantifier (i.e. an instance, or special case, of the generalized wff).
In other words if something is true for all ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1449. Conditional forms of the converse are given by ax12 1512, ax-16 1814, and ax-17 1526.
Unlike the more general textbook Axiom of Specialization, we cannot choose
a variable different from (Contributed by NM, 5-Aug-1993.) |
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Theorem | sp 1511 | Specialization. Another name for ax-4 1510. (Contributed by NM, 21-May-2008.) |
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Theorem | ax12 1512 | Rederive the original version of the axiom from ax-i12 1507. (Contributed by Mario Carneiro, 3-Feb-2015.) |
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Theorem | hbequid 1513 |
Bound-variable hypothesis builder for ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() The proof uses only ax-8 1504 and ax-i12 1507 on top of (the FOL analogue of) modal logic KT. This shows that this can be proved without ax-i9 1530, even though Theorem equid 1701 cannot. A shorter proof using ax-i9 1530 is obtainable from equid 1701 and hbth 1463. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) |
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Theorem | axi12 1514 | Proof that ax-i12 1507 follows from ax-bndl 1509. So that we can track which theorems rely on ax-bndl 1509, proofs should reference ax12or 1508 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.) |
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Theorem | alequcom 1515 |
Commutation law for identical variable specifiers. The antecedent and
consequent are true when ![]() ![]() |
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Theorem | alequcoms 1516 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nalequcoms 1517 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.) |
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Theorem | nfr 1518 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
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Theorem | nfri 1519 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfrd 1520 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | alimd 1521 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | alrimi 1522 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | nfd 1523 |
Deduce that ![]() ![]() |
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Theorem | nfdh 1524 |
Deduce that ![]() ![]() |
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Theorem | nfrimi 1525 |
Moving an antecedent outside ![]() |
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Axiom | ax-17 1526* |
Axiom to quantify a variable over a formula in which it does not occur.
Axiom C5 in [Megill] p. 444 (p. 11 of the
preprint). Also appears as
Axiom B6 (p. 75) of system S2 of [Tarski]
p. 77 and Axiom C5-1 of
[Monk2] p. 113.
(Contributed by NM, 5-Aug-1993.) |
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Theorem | a17d 1527* | ax-17 1526 with antecedent. (Contributed by NM, 1-Mar-2013.) |
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Theorem | nfv 1528* |
If ![]() ![]() ![]() ![]() |
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Theorem | nfvd 1529* | nfv 1528 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1585. (Contributed by Mario Carneiro, 6-Oct-2016.) |
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Axiom | ax-i9 1530 |
Axiom of Existence. One of the equality and substitution axioms of
predicate calculus with equality. One thing this axiom tells us is that
at least one thing exists (although ax-4 1510
and possibly others also tell
us that, i.e. they are not valid in the empty domain of a "free
logic").
In this form (not requiring that ![]() ![]() |
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Theorem | ax-9 1531 | Derive ax-9 1531 from ax-i9 1530, the modified version for intuitionistic logic. Although ax-9 1531 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1530. (Contributed by NM, 3-Feb-2015.) |
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Theorem | equidqe 1532 | equid 1701 with some quantification and negation without using ax-4 1510 or ax-17 1526. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |
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Theorem | ax4sp1 1533 | A special case of ax-4 1510 without using ax-4 1510 or ax-17 1526. (Contributed by NM, 13-Jan-2011.) |
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Axiom | ax-ial 1534 |
![]() ![]() ![]() ![]() |
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Axiom | ax-i5r 1535 | Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.) |
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Theorem | spi 1536 | Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.) |
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Theorem | sps 1537 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | spsd 1538 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
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Theorem | nfbidf 1539 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |
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Theorem | hba1 1540 |
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Theorem | nfa1 1541 |
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Theorem | axc4i 1542 | Inference version of 19.21 1583. (Contributed by NM, 3-Jan-1993.) |
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Theorem | a5i 1543 | Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfnf1 1544 |
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Theorem | hbim 1545 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbor 1546 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hban 1547 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbbi 1548 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hb3or 1549 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hb3an 1550 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hba2 1551 | Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
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Theorem | hbia1 1552 | Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
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Theorem | 19.3h 1553 | A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.) |
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Theorem | 19.3 1554 | A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
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Theorem | 19.16 1555 | Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
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Theorem | 19.17 1556 | Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
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Theorem | 19.21h 1557 |
Theorem 19.21 of [Margaris] p. 90. The
hypothesis can be thought of
as "![]() ![]() |
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Theorem | 19.21bi 1558 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.21bbi 1559 | Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.) |
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Theorem | 19.27h 1560 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.27 1561 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.28h 1562 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.28 1563 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfan1 1564 | A closed form of nfan 1565. (Contributed by Mario Carneiro, 3-Oct-2016.) |
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Theorem | nfan 1565 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nf3an 1566 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nford 1567 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfand 1568 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nf3and 1569 | Deduction form of bound-variable hypothesis builder nf3an 1566. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) |
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Theorem | hbim1 1570 | A closed form of hbim 1545. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfim1 1571 | A closed form of nfim 1572. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
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Theorem | nfim 1572 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbimd 1573 | Deduction form of bound-variable hypothesis builder hbim 1545. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.) |
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Theorem | nfor 1574 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbbid 1575 | Deduction form of bound-variable hypothesis builder hbbi 1548. (Contributed by NM, 1-Jan-2002.) |
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Theorem | nfal 1576 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfnf 1577 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfalt 1578 | Closed form of nfal 1576. (Contributed by Jim Kingdon, 11-May-2018.) |
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Theorem | nfa2 1579 | Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | nfia1 1580 | Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | 19.21ht 1581 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.) |
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Theorem | 19.21t 1582 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) |
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Theorem | 19.21 1583 |
Theorem 19.21 of [Margaris] p. 90. The
hypothesis can be thought of
as "![]() ![]() |
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Theorem | stdpc5 1584 |
An axiom scheme of standard predicate calculus that emulates Axiom 5 of
[Mendelson] p. 69. The hypothesis
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Theorem | nfimd 1585 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | aaanh 1586 | Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) |
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Theorem | aaan 1587 | Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) |
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Theorem | nfbid 1588 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfbi 1589 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 19.8a 1590 | If a wff is true, then it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.8ad 1591 | If a wff is true, it is true for at least one instance. Deduction form of 19.8a 1590. (Contributed by DAW, 13-Feb-2017.) |
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Theorem | 19.23bi 1592 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | exlimih 1593 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
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Theorem | exlimi 1594 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | exlimd2 1595 | Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1596 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.) |
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Theorem | exlimdh 1596 | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.) |
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Theorem | exlimd 1597 | Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.) |
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Theorem | exlimiv 1598* |
Inference from Theorem 19.23 of [Margaris] p.
90.
This inference, along with our many variants is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf. In informal proofs, the statement "Let C be an element such that..." almost always means an implicit application of Rule C.
In essence, Rule C states that if we can prove that some element
We cannot do this in Metamath directly. Instead, we use the original
|
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Theorem | exim 1599 | Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
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Theorem | eximi 1600 | Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
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