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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | sp 1401 | Specialization. Another name for ax-4 1400. (Contributed by NM, 21-May-2008.) |
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Theorem | ax-12 1402 | Rederive the original version of the axiom from ax-i12 1398. (Contributed by Mario Carneiro, 3-Feb-2015.) |
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Theorem | ax12or 1403 | Another name for ax-i12 1398. (Contributed by NM, 3-Feb-2015.) |
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Axiom | ax-13 1404 |
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 1405 |
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 1406 |
Bound-variable hypothesis builder for ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | axi12 1407 | Proof that ax-i12 1398 follows from ax-bndl 1399. So that we can track which theorems rely on ax-bndl 1399, proofs should reference ax-i12 1398 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 1408 |
Commutation law for identical variable specifiers. The antecedent and
consequent are true when ![]() ![]() |
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Theorem | alequcoms 1409 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nalequcoms 1410 | 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 1411 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
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Theorem | nfri 1412 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | nfrd 1413 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
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Theorem | alimd 1414 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | alrimi 1415 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | nfd 1416 |
Deduce that ![]() ![]() |
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Theorem | nfdh 1417 |
Deduce that ![]() ![]() |
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Theorem | nfrimi 1418 |
Moving an antecedent outside ![]() |
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Axiom | ax-17 1419* |
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 1420* | ax-17 1419 with antecedent. (Contributed by NM, 1-Mar-2013.) |
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Theorem | nfv 1421* |
If ![]() ![]() ![]() ![]() |
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Theorem | nfvd 1422* | nfv 1421 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1477. (Contributed by Mario Carneiro, 6-Oct-2016.) |
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Axiom | ax-i9 1423 |
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 1400
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 1424 | Derive ax-9 1424 from ax-i9 1423, the modified version for intuitionistic logic. Although ax-9 1424 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1423. (Contributed by NM, 3-Feb-2015.) |
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Theorem | equidqe 1425 | equid 1589 with some quantification and negation without using ax-4 1400 or ax-17 1419. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |
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Theorem | ax4sp1 1426 | A special case of ax-4 1400 without using ax-4 1400 or ax-17 1419. (Contributed by NM, 13-Jan-2011.) |
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Axiom | ax-ial 1427 |
![]() ![]() ![]() ![]() |
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Axiom | ax-i5r 1428 | Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.) |
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Theorem | spi 1429 | Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.) |
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Theorem | sps 1430 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | spsd 1431 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
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Theorem | nfbidf 1432 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |
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Theorem | hba1 1433 |
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Theorem | nfa1 1434 |
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Theorem | a5i 1435 | Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfnf1 1436 |
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Theorem | hbim 1437 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbor 1438 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hban 1439 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbbi 1440 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hb3or 1441 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hb3an 1442 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hba2 1443 | Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
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Theorem | hbia1 1444 | Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
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Theorem | 19.3h 1445 | 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 1446 | 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 1447 | Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
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Theorem | 19.17 1448 | Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
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Theorem | 19.21h 1449 |
Theorem 19.21 of [Margaris] p. 90. The
hypothesis can be thought of
as "![]() ![]() |
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Theorem | 19.21bi 1450 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.21bbi 1451 | Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.) |
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Theorem | 19.27h 1452 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.27 1453 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.28h 1454 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 19.28 1455 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfan1 1456 | A closed form of nfan 1457. (Contributed by Mario Carneiro, 3-Oct-2016.) |
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Theorem | nfan 1457 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nf3an 1458 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nford 1459 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfand 1460 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nf3and 1461 | Deduction form of bound-variable hypothesis builder nf3an 1458. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) |
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Theorem | hbim1 1462 | A closed form of hbim 1437. (Contributed by NM, 5-Aug-1993.) |
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Theorem | nfim1 1463 | A closed form of nfim 1464. (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 1464 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbimd 1465 | Deduction form of bound-variable hypothesis builder hbim 1437. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.) |
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Theorem | nfor 1466 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | hbbid 1467 | Deduction form of bound-variable hypothesis builder hbbi 1440. (Contributed by NM, 1-Jan-2002.) |
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Theorem | nfal 1468 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfnf 1469 |
If ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfalt 1470 | Closed form of nfal 1468. (Contributed by Jim Kingdon, 11-May-2018.) |
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Theorem | nfa2 1471 | Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | nfia1 1472 | Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | 19.21ht 1473 | 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 1474 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) |
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Theorem | 19.21 1475 |
Theorem 19.21 of [Margaris] p. 90. The
hypothesis can be thought of
as "![]() ![]() |
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Theorem | stdpc5 1476 |
An axiom scheme of standard predicate calculus that emulates Axiom 5 of
[Mendelson] p. 69. The hypothesis
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfimd 1477 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | aaanh 1478 | Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) |
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Theorem | aaan 1479 | Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) |
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Theorem | nfbid 1480 |
If in a context ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | nfbi 1481 |
If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | 19.8a 1482 | If a wff is true, 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 | 19.23bi 1483 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | exlimih 1484 | 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 1485 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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Theorem | exlimd2 1486 | Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1487 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.) |
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Theorem | exlimdh 1487 | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.) |
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Theorem | exlimd 1488 | 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 1489* |
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 1490 | 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 1491 | Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
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Theorem | 2eximi 1492 | Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
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Theorem | eximii 1493 | Inference associated with eximi 1491. (Contributed by BJ, 3-Feb-2018.) |
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Theorem | alinexa 1494 | A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.) |
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Theorem | exbi 1495 | Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
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Theorem | exbii 1496 | Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.) |
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Theorem | 2exbii 1497 | Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.) |
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Theorem | 3exbii 1498 | Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.) |
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Theorem | exancom 1499 | Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.) |
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Theorem | alrimdd 1500 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
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