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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | exbidh 1601 | Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) | ||
Theorem | albid 1602 | Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) | ||
Theorem | exbid 1603 | Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) | ||
Theorem | exsimpl 1604 | Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑) | ||
Theorem | exsimpr 1605 | Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜓) | ||
Theorem | alexdc 1606 | Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1632. (Contributed by Jim Kingdon, 2-Jun-2018.) |
⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)) | ||
Theorem | 19.29 1607 | Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) |
⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | 19.29r 1608 | Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | 19.29r2 1609 | Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.) |
⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓)) | ||
Theorem | 19.29x 1610 | Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.) |
⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓)) | ||
Theorem | 19.35-1 1611 | 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.) |
⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | 19.35i 1612 | Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
⊢ ∃𝑥(𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∃𝑥𝜓) | ||
Theorem | 19.25 1613 | Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∃𝑦∃𝑥𝜓)) | ||
Theorem | 19.30dc 1614 | Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.) |
⊢ (DECID ∃𝑥𝜓 → (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))) | ||
Theorem | 19.43 1615 | Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) | ||
Theorem | 19.33b2 1616 | The antecedent provides a condition implying the converse of 19.33 1471. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1617 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.) |
⊢ ((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))) | ||
Theorem | 19.33bdc 1617 | Converse of 19.33 1471 given ¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) 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 1616 (Contributed by Jim Kingdon, 23-Apr-2018.) |
⊢ (DECID ∃𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))) | ||
Theorem | 19.40 1618 | Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓)) | ||
Theorem | 19.40-2 1619 | Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓)) | ||
Theorem | exintrbi 1620 | Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ 𝜓))) | ||
Theorem | exintr 1621 | Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) | ||
Theorem | alsyl 1622 | Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.) |
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜒)) → ∀𝑥(𝜑 → 𝜒)) | ||
Theorem | hbex 1623 | If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) | ||
Theorem | nfex 1624 | If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥∃𝑦𝜑 | ||
Theorem | 19.2 1625 | Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.) |
⊢ (∀𝑥𝜑 → ∃𝑦𝜑) | ||
Theorem | i19.24 1626 | Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1611, 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.) |
⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) ⇒ ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) | ||
Theorem | i19.39 1627 | Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1611, 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.) |
⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) ⇒ ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) | ||
Theorem | 19.9ht 1628 | A closed version of one direction of 19.9 1631. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | ||
Theorem | 19.9t 1629 | A closed version of 19.9 1631. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.) |
⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑)) | ||
Theorem | 19.9h 1630 | 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.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃𝑥𝜑 ↔ 𝜑) | ||
Theorem | 19.9 1631 | 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.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥𝜑 ↔ 𝜑) | ||
Theorem | alexim 1632 | One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1606. (Contributed by Jim Kingdon, 2-Jul-2018.) |
⊢ (∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑) | ||
Theorem | exnalim 1633 | One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.) |
⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑) | ||
Theorem | exanaliim 1634 | A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.) |
⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | alexnim 1635 | A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.) |
⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ¬ ∃𝑥∀𝑦𝜑) | ||
Theorem | nnal 1636 | The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.) |
⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑) | ||
Theorem | ax6blem 1637 | If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1638 compared to hbnt 1640. (Contributed by GD, 27-Jan-2018.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) | ||
Theorem | ax6b 1638 |
Quantified Negation. Axiom C5-2 of [Monk2] p.
113.
(Contributed by GD, 27-Jan-2018.) |
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hbn1 1639 | 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.) |
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | ||
Theorem | hbnt 1640 | Closed theorem version of bound-variable hypothesis builder hbn 1641. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑)) | ||
Theorem | hbn 1641 | If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) | ||
Theorem | hbnd 1642 | Deduction form of bound-variable hypothesis builder hbn 1641. (Contributed by NM, 3-Jan-2002.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓)) | ||
Theorem | nfnt 1643 | If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) |
⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑) | ||
Theorem | nfnd 1644 | Deduction associated with nfnt 1643. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) | ||
Theorem | nfn 1645 | Inference associated with nfnt 1643. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 ¬ 𝜑 | ||
Theorem | nfdc 1646 | If 𝑥 is not free in 𝜑, it is not free in DECID 𝜑. (Contributed by Jim Kingdon, 11-Mar-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥DECID 𝜑 | ||
Theorem | modal-5 1647 | The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.) |
⊢ (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑) | ||
Theorem | 19.9d 1648 | A deduction version of one direction of 19.9 1631. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ (𝜓 → Ⅎ𝑥𝜑) ⇒ ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑)) | ||
Theorem | 19.9hd 1649 | A deduction version of one direction of 19.9 1631. This is an older variation of this theorem; new proofs should use 19.9d 1648. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝜓 → (𝜑 → ∀𝑥𝜑)) ⇒ ⊢ (𝜓 → (∃𝑥𝜑 → 𝜑)) | ||
Theorem | excomim 1650 | One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑) | ||
Theorem | excom 1651 | Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) | ||
Theorem | 19.12 1652 | Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.) |
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) | ||
Theorem | 19.19 1653 | Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓)) | ||
Theorem | 19.21-2 1654 | Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 ⇒ ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓)) | ||
Theorem | nf2 1655 | An alternate definition of df-nf 1448, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | ||
Theorem | nf3 1656 | An alternate definition of df-nf 1448. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑)) | ||
Theorem | nf4dc 1657 | Variable 𝑥 is effectively not free in 𝜑 iff 𝜑 is always true or always false, given a decidability condition. The reverse direction, nf4r 1658, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.) |
⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))) | ||
Theorem | nf4r 1658 | If 𝜑 is always true or always false, then variable 𝑥 is effectively not free in 𝜑. The converse holds given a decidability condition, as seen at nf4dc 1657. (Contributed by Jim Kingdon, 21-Jul-2018.) |
⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑) | ||
Theorem | 19.36i 1659 | Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
⊢ Ⅎ𝑥𝜓 & ⊢ ∃𝑥(𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | 19.36-1 1660 | Closed form of 19.36i 1659. 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.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | 19.37-1 1661 | 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.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) | ||
Theorem | 19.37aiv 1662* | Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ∃𝑥(𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | 19.38 1663 | Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)) | ||
Theorem | 19.23t 1664 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ (Ⅎ𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))) | ||
Theorem | 19.23 1665 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)) | ||
Theorem | 19.32dc 1666 | Theorem 19.32 of [Margaris] p. 90, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))) | ||
Theorem | 19.32r 1667 | One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1666. (Contributed by Jim Kingdon, 28-Jul-2018.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | 19.31r 1668 | 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.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ ((∀𝑥𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | 19.44 1669 | Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓)) | ||
Theorem | 19.45 1670 | Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓)) | ||
Theorem | 19.34 1671 | Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | 19.41h 1672 | Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1673 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.41 1673 | 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.) |
⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓)) | ||
Theorem | 19.42h 1674 | Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1675 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) | ||
Theorem | 19.42 1675 | Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)) | ||
Theorem | excom13 1676 | Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑) | ||
Theorem | exrot3 1677 | Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑) | ||
Theorem | exrot4 1678 | Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.) |
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑) | ||
Theorem | nexr 1679 | Inference from 19.8a 1577. (Contributed by Jeff Hankins, 26-Jul-2009.) |
⊢ ¬ ∃𝑥𝜑 ⇒ ⊢ ¬ 𝜑 | ||
Theorem | exan 1680 | Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (∃𝑥𝜑 ∧ 𝜓) ⇒ ⊢ ∃𝑥(𝜑 ∧ 𝜓) | ||
Theorem | hbexd 1681 | Deduction form of bound-variable hypothesis builder hbex 1623. (Contributed by NM, 2-Jan-2002.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) ⇒ ⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓)) | ||
Theorem | eeor 1682 | Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓)) | ||
Theorem | a9e 1683 | 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 1434 through ax-14 2138 and ax-17 1513, all axioms other than ax-9 1518 are believed to be theorems of free logic, although the system without ax-9 1518 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
⊢ ∃𝑥 𝑥 = 𝑦 | ||
Theorem | a9ev 1684* | At least one individual exists. Weaker version of a9e 1683. (Contributed by NM, 3-Aug-2017.) |
⊢ ∃𝑥 𝑥 = 𝑦 | ||
Theorem | ax9o 1685 | An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) | ||
Theorem | spimfv 1686* | Specialization, using implicit substitution. Version of spim 1725 with a disjoint variable condition. See spimv 1798 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | chvarfv 1687* | Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 1744 with a disjoint variable condition. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | equid 1688 |
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 𝑦. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.) |
⊢ 𝑥 = 𝑥 | ||
Theorem | nfequid 1689 | Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.) |
⊢ Ⅎ𝑦 𝑥 = 𝑥 | ||
Theorem | stdpc6 1690 | One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1757.) 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.) |
⊢ ∀𝑥 𝑥 = 𝑥 | ||
Theorem | equcomi 1691 | Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥) | ||
Theorem | ax6evr 1692* | A commuted form of a9ev 1684. 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.) |
⊢ ∃𝑥 𝑦 = 𝑥 | ||
Theorem | equcom 1693 | Commutative law for equality. (Contributed by NM, 20-Aug-1993.) |
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | ||
Theorem | equcomd 1694 | Deduction form of equcom 1693, symmetry of equality. For the versions for classes, see eqcom 2166 and eqcomd 2170. (Contributed by BJ, 6-Oct-2019.) |
⊢ (𝜑 → 𝑥 = 𝑦) ⇒ ⊢ (𝜑 → 𝑦 = 𝑥) | ||
Theorem | equcoms 1695 | An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (𝑦 = 𝑥 → 𝜑) | ||
Theorem | equtr 1696 | A transitive law for equality. (Contributed by NM, 23-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | ||
Theorem | equtrr 1697 | A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦)) | ||
Theorem | equtr2 1698 | A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) | ||
Theorem | equequ1 1699 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | ||
Theorem | equequ2 1700 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦)) |
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