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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | axi12 1501 | Proof that ax-i12 1494 follows from ax-bndl 1496. So that we can track which theorems rely on ax-bndl 1496, proofs should reference ax12or 1495 rather than this theorem. (Contributed by Jim Kingdon, 17-Aug-2018.) (New usage is discouraged). (Proof modification is discouraged.) |
⊢ (∀𝑧 𝑧 = 𝑥 ∨ (∀𝑧 𝑧 = 𝑦 ∨ ∀𝑧(𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))) | ||
Theorem | alequcom 1502 | Commutation law for identical variable specifiers. The antecedent and consequent are true when 𝑥 and 𝑦 are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | alequcoms 1503 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | nalequcoms 1504 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | nfr 1505 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | ||
Theorem | nfri 1506 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | nfrd 1507 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | alimd 1508 | Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) | ||
Theorem | alrimi 1509 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜓) | ||
Theorem | nfd 1510 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Theorem | nfdh 1511 | Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Theorem | nfrimi 1512 | Moving an antecedent outside Ⅎ. (Contributed by Jim Kingdon, 23-Mar-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥(𝜑 → 𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Axiom | ax-17 1513* |
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.) |
⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | a17d 1514* | ax-17 1513 with antecedent. (Contributed by NM, 1-Mar-2013.) |
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | ||
Theorem | nfv 1515* | If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 | ||
Theorem | nfvd 1516* | nfv 1515 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1572. (Contributed by Mario Carneiro, 6-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) | ||
Axiom | ax-i9 1517 | 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 1497 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 𝑥 and 𝑦 be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) Another name for this theorem is a9e 1683, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ ∃𝑥 𝑥 = 𝑦 | ||
Theorem | ax-9 1518 | Derive ax-9 1518 from ax-i9 1517, the modified version for intuitionistic logic. Although ax-9 1518 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1517. (Contributed by NM, 3-Feb-2015.) |
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | ||
Theorem | equidqe 1519 | equid 1688 with some quantification and negation without using ax-4 1497 or ax-17 1513. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |
⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥 | ||
Theorem | ax4sp1 1520 | A special case of ax-4 1497 without using ax-4 1497 or ax-17 1513. (Contributed by NM, 13-Jan-2011.) |
⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥) | ||
Axiom | ax-ial 1521 | 𝑥 is not free in ∀𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Axiom | ax-i5r 1522 | Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)) | ||
Theorem | spi 1523 | Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.) |
⊢ ∀𝑥𝜑 ⇒ ⊢ 𝜑 | ||
Theorem | sps 1524 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spsd 1525 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
Theorem | nfbidf 1526 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒)) | ||
Theorem | hba1 1527 | 𝑥 is not free in ∀𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | ||
Theorem | nfa1 1528 | 𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥∀𝑥𝜑 | ||
Theorem | axc4i 1529 | Inference version of 19.21 1570. (Contributed by NM, 3-Jan-1993.) |
⊢ (∀𝑥𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | a5i 1530 | Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | nfnf1 1531 | 𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥Ⅎ𝑥𝜑 | ||
Theorem | hbim 1532 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | hbor 1533 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | hban 1534 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | hbbi 1535 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ↔ 𝜓). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ ((𝜑 ↔ 𝜓) → ∀𝑥(𝜑 ↔ 𝜓)) | ||
Theorem | hb3or 1536 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∨ 𝜓 ∨ 𝜒). (Contributed by NM, 14-Sep-2003.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → ∀𝑥(𝜑 ∨ 𝜓 ∨ 𝜒)) | ||
Theorem | hb3an 1537 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by NM, 14-Sep-2003.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜒 → ∀𝑥𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | hba2 1538 | Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦∀𝑥𝜑) | ||
Theorem | hbia1 1539 | Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | 19.3h 1540 | 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.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.3 1541 | 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.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.16 1542 | Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓)) | ||
Theorem | 19.17 1543 | Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓)) | ||
Theorem | 19.21h 1544 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." New proofs should use 19.21 1570 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | ||
Theorem | 19.21bi 1545 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | 19.21bbi 1546 | Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.) |
⊢ (𝜑 → ∀𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | 19.27h 1547 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.27 1548 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.28h 1549 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | 19.28 1550 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | nfan1 1551 | A closed form of nfan 1552. (Contributed by Mario Carneiro, 3-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) | ||
Theorem | nfan 1552 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) | ||
Theorem | nf3an 1553 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) | ||
Theorem | nford 1554 | If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓 ∨ 𝜒). (Contributed by Jim Kingdon, 29-Oct-2019.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∨ 𝜒)) | ||
Theorem | nfand 1555 | If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒)) | ||
Theorem | nf3and 1556 | Deduction form of bound-variable hypothesis builder nf3an 1553. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → Ⅎ𝑥𝜃) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | hbim1 1557 | A closed form of hbim 1532. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | nfim1 1558 | A closed form of nfim 1559. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ Ⅎ𝑥(𝜑 → 𝜓) | ||
Theorem | nfim 1559 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 → 𝜓) | ||
Theorem | hbimd 1560 | Deduction form of bound-variable hypothesis builder hbim 1532. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) | ||
Theorem | nfor 1561 | If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by Jim Kingdon, 11-Mar-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) | ||
Theorem | hbbid 1562 | Deduction form of bound-variable hypothesis builder hbbi 1535. (Contributed by NM, 1-Jan-2002.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ↔ 𝜒) → ∀𝑥(𝜓 ↔ 𝜒))) | ||
Theorem | nfal 1563 | If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-4 1497. (Revised by Gino Giotto, 25-Aug-2024.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∀𝑦𝜑 | ||
Theorem | nfnf 1564 | If 𝑥 is not free in 𝜑, it is not free in Ⅎ𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥Ⅎ𝑦𝜑 | ||
Theorem | nfalt 1565 | Closed form of nfal 1563. (Contributed by Jim Kingdon, 11-May-2018.) |
⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥∀𝑦𝜑) | ||
Theorem | nfa2 1566 | Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑 | ||
Theorem | nfia1 1567 | Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥(∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | 19.21ht 1568 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
Theorem | 19.21t 1569 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) |
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
Theorem | 19.21 1570 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | ||
Theorem | stdpc5 1571 | An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎ𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 1689. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)) | ||
Theorem | nfimd 1572 | If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒)) | ||
Theorem | aaanh 1573 | Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) | ||
Theorem | aaan 1574 | Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) | ||
Theorem | nfbid 1575 | If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ↔ 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒)) | ||
Theorem | nfbi 1576 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ↔ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ↔ 𝜓) | ||
Theorem | 19.8a 1577 | 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.) |
⊢ (𝜑 → ∃𝑥𝜑) | ||
Theorem | 19.8ad 1578 | If a wff is true, it is true for at least one instance. Deduction form of 19.8a 1577. (Contributed by DAW, 13-Feb-2017.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | 19.23bi 1579 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃𝑥𝜑 → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | exlimih 1580 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥𝜑 → 𝜓) | ||
Theorem | exlimi 1581 | Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥𝜑 → 𝜓) | ||
Theorem | exlimd2 1582 | Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1583 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) | ||
Theorem | exlimdh 1583 | Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜒 → ∀𝑥𝜒) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) | ||
Theorem | exlimd 1584 | Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) | ||
Theorem | exlimiv 1585* |
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 𝑥 exists satisfying a wff, i.e. ∃𝑥𝜑(𝑥) where 𝜑(𝑥) has 𝑥 free, then we can use 𝜑( C ) as a hypothesis for the proof where C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier. We cannot do this in Metamath directly. Instead, we use the original 𝜑 (containing 𝑥) as an antecedent for the main part of the proof. We eventually arrive at (𝜑 → 𝜓) where 𝜓 is the theorem to be proved and does not contain 𝑥. Then we apply exlimiv 1585 to arrive at (∃𝑥𝜑 → 𝜓). Finally, we separately prove ∃𝑥𝜑 and detach it with modus ponens ax-mp 5 to arrive at the final theorem 𝜓. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 25-Jul-2012.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥𝜑 → 𝜓) | ||
Theorem | exim 1586 | Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | eximi 1587 | Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑥𝜓) | ||
Theorem | 2eximi 1588 | Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓) | ||
Theorem | eximii 1589 | Inference associated with eximi 1587. (Contributed by BJ, 3-Feb-2018.) |
⊢ ∃𝑥𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ ∃𝑥𝜓 | ||
Theorem | alinexa 1590 | A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | exbi 1591 | Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓)) | ||
Theorem | exbii 1592 | Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑥𝜓) | ||
Theorem | 2exbii 1593 | Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓) | ||
Theorem | 3exbii 1594 | Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓) | ||
Theorem | exancom 1595 | Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.) |
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) | ||
Theorem | alrimdd 1596 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | ||
Theorem | alrimd 1597 | Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | ||
Theorem | eximdh 1598 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | eximd 1599 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | nexd 1600 | Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → ¬ ∃𝑥𝜓) |
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