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Type | Label | Description |
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Statement | ||
Theorem | alim 1401 | Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.) |
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Theorem | al2imi 1402 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alanimi 1403 | Variant of al2imi 1402 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
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Syntax | wnf 1404 | Extend wff definition to include the not-free predicate. |
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Definition | df-nf 1405 |
Define the not-free predicate for wffs. This is read "![]() ![]() ![]() ![]() ![]() ![]() Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.
To be precise, our definition really means "effectively not
free," because
it is slightly less restrictive than the usual textbook definition for
not-free (which only considers syntactic freedom). For example, |
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Theorem | nfi 1406 |
Deduce that ![]() ![]() |
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Theorem | hbth 1407 |
No variable is (effectively) free in a theorem.
This and later "hypothesis-building" lemmas, with labels
starting
"hb...", allow us to construct proofs of formulas of the form
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Theorem | nfth 1408 | No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfnth 1409 | No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) |
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Theorem | nftru 1410 | The true constant has no free variables. (This can also be proven in one step with nfv 1476, but this proof does not use ax-17 1474.) (Contributed by Mario Carneiro, 6-Oct-2016.) |
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Theorem | alimdh 1411 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.) |
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Theorem | albi 1412 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alrimih 1413 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
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Theorem | albii 1414 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
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Theorem | 2albii 1415 | Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.) |
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Theorem | hbxfrbi 1416 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
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Theorem | nfbii 1417 | Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfxfr 1418 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfxfrd 1419 | A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | alcoms 1420 | Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.) |
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Theorem | hbal 1421 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | alcom 1422 | Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alrimdh 1423 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
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Theorem | albidh 1424 | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.26 1425 | Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
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Theorem | 19.26-2 1426 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
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Theorem | 19.26-3an 1427 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
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Theorem | 19.33 1428 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | alrot3 1429 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
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Theorem | alrot4 1430 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
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Theorem | albiim 1431 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
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Theorem | 2albiim 1432 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
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Theorem | hband 1433 | Deduction form of bound-variable hypothesis builder hban 1494. (Contributed by NM, 2-Jan-2002.) |
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Theorem | hb3and 1434 | Deduction form of bound-variable hypothesis builder hb3an 1497. (Contributed by NM, 17-Feb-2013.) |
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Theorem | hbald 1435 | Deduction form of bound-variable hypothesis builder hbal 1421. (Contributed by NM, 2-Jan-2002.) |
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Syntax | wex 1436 | Extend wff definition to include the existential quantifier ("there exists"). |
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Axiom | ax-ie1 1437 |
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Axiom | ax-ie2 1438 |
Define existential quantification. ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbe1 1439 |
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Theorem | nfe1 1440 |
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Theorem | 19.23ht 1441 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.) |
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Theorem | 19.23h 1442 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.) |
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Theorem | alnex 1443 |
Theorem 19.7 of [Margaris] p. 89. To read
this intuitionistically, think
of it as "if ![]() ![]() ![]() ![]() |
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Theorem | nex 1444 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
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Theorem | dfexdc 1445 |
Defining ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | exalim 1446 | 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 1445. (Contributed by Jim Kingdon, 29-Jul-2018.) |
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The equality predicate was introduced above in wceq 1299 for use by df-tru 1302. See the comments in that section. In this section, we continue with the first "real" use of it. | ||
Theorem | weq 1447 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1447 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 1299. This lets us avoid overloading
the |
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Syntax | wcel 1448 |
Extend wff definition to include the membership connective between
classes.
(The purpose of introducing |
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Theorem | wel 1449 |
Extend wff definition to include atomic formulas with the epsilon
(membership) predicate. This is read "![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
This syntactical construction introduces a binary non-logical predicate
symbol
(Instead of introducing wel 1449 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 wcel 1448. This lets us avoid overloading
the |
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Axiom | ax-8 1450 |
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 1653). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1450 through ax-16 1753 are the axioms having to do with equality,
substitution, and logical properties of our binary predicate |
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Axiom | ax-10 1451 |
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 1662 ("o" for "old") and was replaced with this shorter ax-10 1451 in May 2008. The old axiom is proved from this one as theorem ax10o 1661. Conversely, this axiom is proved from ax-10o 1662 as theorem ax10 1663. (Contributed by NM, 5-Aug-1993.) |
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Axiom | ax-11 1452 |
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 1766, ax11v2 1759 and ax-11o 1762. (Contributed by NM, 5-Aug-1993.) |
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Axiom | ax-i12 1453 |
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 ax-12 1457 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
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Axiom | ax-bndl 1454 |
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 1453 as can be seen at axi12 1462. Whether ax-bndl 1454 can be proved from the remaining axioms including ax-i12 1453 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 1455 |
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 1393. Conditional forms of the converse are given by ax-12 1457, ax-16 1753, and ax-17 1474.
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 1456 | Specialization. Another name for ax-4 1455. (Contributed by NM, 21-May-2008.) |
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Theorem | ax-12 1457 | Rederive the original version of the axiom from ax-i12 1453. (Contributed by Mario Carneiro, 3-Feb-2015.) |
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Theorem | ax12or 1458 | Another name for ax-i12 1453. (Contributed by NM, 3-Feb-2015.) |
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Axiom | ax-13 1459 |
Axiom of Equality. One of the equality and substitution axioms for a
non-logical predicate in our predicate calculus with equality. It
substitutes equal variables into the left-hand side of the ![]() |
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Axiom | ax-14 1460 |
Axiom of Equality. One of the equality and substitution axioms for a
non-logical predicate in our predicate calculus with equality. It
substitutes equal variables into the right-hand side of the ![]() |
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Theorem | hbequid 1461 |
Bound-variable hypothesis builder for ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | axi12 1462 | Proof that ax-i12 1453 follows from ax-bndl 1454. So that we can track which theorems rely on ax-bndl 1454, proofs should reference ax-i12 1453 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 1463 |
Commutation law for identical variable specifiers. The antecedent and
consequent are true when ![]() ![]() |
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Theorem | alequcoms 1464 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nalequcoms 1465 | 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 1466 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
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Theorem | nfri 1467 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfrd 1468 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | alimd 1469 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | alrimi 1470 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | nfd 1471 |
Deduce that ![]() ![]() |
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Theorem | nfdh 1472 |
Deduce that ![]() ![]() |
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Theorem | nfrimi 1473 |
Moving an antecedent outside ![]() |
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Axiom | ax-17 1474* |
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 1475* | ax-17 1474 with antecedent. (Contributed by NM, 1-Mar-2013.) |
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Theorem | nfv 1476* |
If ![]() ![]() ![]() ![]() |
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Theorem | nfvd 1477* | nfv 1476 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1532. (Contributed by Mario Carneiro, 6-Oct-2016.) |
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Axiom | ax-i9 1478 |
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 1455
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 1479 | Derive ax-9 1479 from ax-i9 1478, the modified version for intuitionistic logic. Although ax-9 1479 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1478. (Contributed by NM, 3-Feb-2015.) |
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Theorem | equidqe 1480 | equid 1645 with some quantification and negation without using ax-4 1455 or ax-17 1474. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |
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Theorem | ax4sp1 1481 | A special case of ax-4 1455 without using ax-4 1455 or ax-17 1474. (Contributed by NM, 13-Jan-2011.) |
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Axiom | ax-ial 1482 |
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Axiom | ax-i5r 1483 | Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.) |
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Theorem | spi 1484 | Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.) |
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Theorem | sps 1485 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | spsd 1486 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
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Theorem | nfbidf 1487 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |
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Theorem | hba1 1488 |
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Theorem | nfa1 1489 |
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Theorem | a5i 1490 | Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfnf1 1491 |
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Theorem | hbim 1492 |
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Theorem | hbor 1493 |
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Theorem | hban 1494 |
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Theorem | hbbi 1495 |
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Theorem | hb3or 1496 |
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Theorem | hb3an 1497 |
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Theorem | hba2 1498 | Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
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Theorem | hbia1 1499 | Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
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Theorem | 19.3h 1500 | 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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