Home | Intuitionistic Logic Explorer Theorem List (p. 18 of 141) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | equcoms 1701 | 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 1702 | A transitive law for equality. (Contributed by NM, 23-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | ||
Theorem | equtrr 1703 | 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 1704 | 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 1705 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧)) | ||
Theorem | equequ2 1706 | An equivalence law for equality. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦)) | ||
Theorem | ax11i 1707 | Inference that has ax-11 1499 (without ∀𝑦) as its conclusion and does not require ax-10 1498, ax-11 1499, or ax12 1505 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | ||
Theorem | ax10o 1708 |
Show that ax-10o 1709 can be derived from ax-10 1498. An open problem is
whether this theorem can be derived from ax-10 1498 and the others when
ax-11 1499 is replaced with ax-11o 1816. See Theorem ax10 1710
for the
rederivation of ax-10 1498 from ax10o 1708.
Normally, ax10o 1708 should be used rather than ax-10o 1709, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Axiom | ax-10o 1709 |
Axiom ax-10o 1709 ("o" for "old") was the
original version of ax-10 1498,
before it was discovered (in May 2008) that the shorter ax-10 1498 could
replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of
the preprint).
This axiom is redundant, as shown by Theorem ax10o 1708. Normally, ax10o 1708 should be used rather than ax-10o 1709, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) | ||
Theorem | ax10 1710 |
Rederivation of ax-10 1498 from original version ax-10o 1709. See Theorem
ax10o 1708 for the derivation of ax-10o 1709 from ax-10 1498.
This theorem should not be referenced in any proof. Instead, use ax-10 1498 above so that uses of ax-10 1498 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | ||
Theorem | hbae 1711 | All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑧∀𝑥 𝑥 = 𝑦) | ||
Theorem | nfae 1712 | All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦 | ||
Theorem | hbaes 1713 | Rule that applies hbae 1711 to antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑧∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → 𝜑) | ||
Theorem | hbnae 1714 | All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | nfnae 1715 | All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦 | ||
Theorem | hbnaes 1716 | Rule that applies hbnae 1714 to antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) | ||
Theorem | naecoms 1717 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) ⇒ ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) | ||
Theorem | equs4 1718 | Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | equsalh 1719 | A useful equivalence related to substitution. New proofs should use equsal 1720 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | equsal 1720 | A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | equsex 1721 | A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) | ||
Theorem | equsexd 1722 | Deduction form of equsex 1721. (Contributed by Jim Kingdon, 29-Dec-2017.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒)) | ||
Theorem | dral1 1723 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓)) | ||
Theorem | dral2 1724 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓)) | ||
Theorem | drex2 1725 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 ↔ ∃𝑧𝜓)) | ||
Theorem | drnf1 1726 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓)) | ||
Theorem | drnf2 1727 | Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓)) | ||
Theorem | spimth 1728 | Closed theorem form of spim 1731. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.) |
⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spimt 1729 | Closed theorem form of spim 1731. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.) |
⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) | ||
Theorem | spimh 1730 | Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1731 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 8-May-2008.) (New usage is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spim 1731 | Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1731 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → 𝜓) | ||
Theorem | spimeh 1732 | Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | spimed 1733 | Deduction version of spime 1734. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) |
⊢ (𝜒 → Ⅎ𝑥𝜑) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) | ||
Theorem | spime 1734 | Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (𝜑 → ∃𝑥𝜓) | ||
Theorem | cbv3 1735 | Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because proofs are encouraged to use the weaker cbv3v 1737 if possible. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | cbv3h 1736 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | cbv3v 1737* | Rule used to change bound variables, using implicit substitution. Version of cbv3 1735 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑦𝜓) | ||
Theorem | cbv1 1738 | Rule used to change bound variables, using implicit substitution. Revised to format hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) | ||
Theorem | cbv1h 1739 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.) |
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) | ||
Theorem | cbv1v 1740* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 16-Jun-2019.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒)) | ||
Theorem | cbv2h 1741 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 → (𝜓 → ∀𝑦𝜓)) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | cbv2 1742 | Rule used to change bound variables, using implicit substitution. Revised to align format of hypotheses to common style. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | cbv2w 1743* | Rule used to change bound variables, using implicit substitution. Version of cbv2 1742 with a disjoint variable condition. (Contributed by NM, 5-Aug-1993.) (Revised by Gino Giotto, 10-Jan-2024.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑦𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) | ||
Theorem | cbvalv1 1744* | Rule used to change bound variables, using implicit substitution. Version of cbval 1747 with a disjoint variable condition. See cbvalvw 1912 for a version with two disjoint variable conditions, and cbvalv 1910 for another variant. (Contributed by NM, 13-May-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbvexv1 1745* | Rule used to change bound variables, using implicit substitution. Version of cbvex 1749 with a disjoint variable condition. See cbvexvw 1913 for a version with two disjoint variable conditions, and cbvexv 1911 for another variant. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Theorem | cbvalh 1746 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbval 1747 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓) | ||
Theorem | cbvexh 1748 | Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.) |
⊢ (𝜑 → ∀𝑦𝜑) & ⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Theorem | cbvex 1749 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) | ||
Theorem | chvar 1750 | Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ 𝜑 ⇒ ⊢ 𝜓 | ||
Theorem | equvini 1751 | A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦 (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) | ||
Theorem | equveli 1752 | A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1751.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.) |
⊢ (∀𝑧(𝑧 = 𝑥 ↔ 𝑧 = 𝑦) → 𝑥 = 𝑦) | ||
Theorem | nfald 1753 | If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓) | ||
Theorem | nfexd 1754 | If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.) |
⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) | ||
Syntax | wsb 1755 | Extend wff definition to include proper substitution (read "the wff that results when 𝑦 is properly substituted for 𝑥 in wff 𝜑"). (Contributed by NM, 24-Jan-2006.) |
wff [𝑦 / 𝑥]𝜑 | ||
Definition | df-sb 1756 |
Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the
preprint). For our notation, we use [𝑦 / 𝑥]𝜑 to mean "the wff
that results when 𝑦 is properly substituted for 𝑥 in the
wff
𝜑". We can also use [𝑦 / 𝑥]𝜑 in place of the "free for"
side condition used in traditional predicate calculus; see, for example,
stdpc4 1768.
Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑦) is the wff that results when 𝑦 is properly substituted for 𝑥 in 𝜑(𝑥)". For example, if the original 𝜑(𝑥) is 𝑥 = 𝑦, then 𝜑(𝑦) is 𝑦 = 𝑦, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see Theorems sbequ 1833, sbcom2 1980 and sbid2v 1989). Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 1767 shows. We achieve this by having 𝑥 free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 1984 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another alternate definition which uses a dummy variable is dfsb7a 1987. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 1880 and sb6 1879. In classical logic, another possible definition is (𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑) but we do not have an intuitionistic proof that this is equivalent. There are no restrictions on any of the variables, including what variables may occur in wff 𝜑. (Contributed by NM, 5-Aug-1993.) |
⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | ||
Theorem | sbimi 1757 | Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) | ||
Theorem | sbbii 1758 | Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓) | ||
Theorem | sb1 1759 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | sb2 1760 | One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | ||
Theorem | sbequ1 1761 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | ||
Theorem | sbequ2 1762 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑)) | ||
Theorem | stdpc7 1763 | One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1696.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑦)". Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 → 𝜑)) | ||
Theorem | sbequ12 1764 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | ||
Theorem | sbequ12r 1765 | An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑)) | ||
Theorem | sbequ12a 1766 | An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) |
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)) | ||
Theorem | sbid 1767 | An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | stdpc4 1768 | The specialization axiom of standard predicate calculus. It states that if a statement 𝜑 holds for all 𝑥, then it also holds for the specific case of 𝑦 (properly) substituted for 𝑥. Translated to traditional notation, it can be read: "∀𝑥𝜑(𝑥) → 𝜑(𝑦), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥)". Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | ||
Theorem | sbh 1769 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | sbf 1770 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) | ||
Theorem | sbf2 1771 | Substitution has no effect on a bound variable. (Contributed by NM, 1-Jul-2005.) |
⊢ ([𝑦 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑) | ||
Theorem | sb6x 1772 | Equivalence involving substitution for a variable not free. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | nfs1f 1773 | If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | ||
Theorem | hbs1f 1774 | If 𝑥 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (𝜑 → ∀𝑥𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | sbequ5 1775 | Substitution does not change an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-Dec-2004.) |
⊢ ([𝑤 / 𝑧]∀𝑥 𝑥 = 𝑦 ↔ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | sbequ6 1776 | Substitution does not change a distinctor. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 14-May-2005.) |
⊢ ([𝑤 / 𝑧] ¬ ∀𝑥 𝑥 = 𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦) | ||
Theorem | sbt 1777 | A substitution into a theorem remains true. (See chvar 1750 and chvarv 1930 for versions using implicit substitition.) (Contributed by NM, 21-Jan-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ 𝜑 ⇒ ⊢ [𝑦 / 𝑥]𝜑 | ||
Theorem | equsb1 1778 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
⊢ [𝑦 / 𝑥]𝑥 = 𝑦 | ||
Theorem | equsb2 1779 | Substitution applied to an atomic wff. (Contributed by NM, 5-Aug-1993.) |
⊢ [𝑦 / 𝑥]𝑦 = 𝑥 | ||
Theorem | sbiedh 1780 | Conversion of implicit substitution to explicit substitution (deduction version of sbieh 1783). New proofs should use sbied 1781 instead. (Contributed by NM, 30-Jun-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | sbied 1781 | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1784). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Ⅎ𝑥𝜒) & ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | sbiedv 1782* | Conversion of implicit substitution to explicit substitution (deduction version of sbie 1784). (Contributed by NM, 7-Jan-2017.) |
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒)) | ||
Theorem | sbieh 1783 | Conversion of implicit substitution to explicit substitution. New proofs should use sbie 1784 instead. (Contributed by NM, 30-Jun-1994.) (New usage is discouraged.) |
⊢ (𝜓 → ∀𝑥𝜓) & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbie 1784 | Conversion of implicit substitution to explicit substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Revised by Wolf Lammen, 30-Apr-2018.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | sbiev 1785* | Conversion of implicit substitution to explicit substitution. Version of sbie 1784 with a disjoint variable condition. (Contributed by Wolf Lammen, 18-Jan-2023.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | ||
Theorem | equsalv 1786* | An equivalence related to implicit substitution. Version of equsal 1720 with a disjoint variable condition. (Contributed by NM, 2-Jun-1993.) (Revised by BJ, 31-May-2019.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) | ||
Theorem | equs5a 1787 | A property related to substitution that unlike equs5 1822 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | equs5e 1788 | A property related to substitution that unlike equs5 1822 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Revised by NM, 3-Feb-2015.) |
⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | ||
Theorem | ax11e 1789 | Analogue to ax-11 1499 but for existential quantification. (Contributed by Mario Carneiro and Jim Kingdon, 31-Dec-2017.) (Proved by Mario Carneiro, 9-Feb-2018.) |
⊢ (𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑦𝜑)) | ||
Theorem | ax10oe 1790 | Quantifier Substitution for existential quantifiers. Analogue to ax10o 1708 but for ∃ rather than ∀. (Contributed by Jim Kingdon, 21-Dec-2017.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜓 → ∃𝑦𝜓)) | ||
Theorem | drex1 1791 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) (Revised by NM, 3-Feb-2015.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝜑 ↔ ∃𝑦𝜓)) | ||
Theorem | drsb1 1792 | Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜑)) | ||
Theorem | exdistrfor 1793 | Distribution of existential quantifiers, with a bound-variable hypothesis saying that 𝑦 is not free in 𝜑, but 𝑥 can be free in 𝜑 (and there is no distinct variable condition on 𝑥 and 𝑦). (Contributed by Jim Kingdon, 25-Feb-2018.) |
⊢ (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥Ⅎ𝑦𝜑) ⇒ ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ ∃𝑦𝜓)) | ||
Theorem | sb4a 1794 | A version of sb4 1825 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | equs45f 1795 | Two ways of expressing substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 25-Apr-2008.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb6f 1796 | Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 30-Apr-2008.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | ||
Theorem | sb5f 1797 | Equivalence for substitution when 𝑦 is not free in 𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 18-May-2008.) |
⊢ (𝜑 → ∀𝑦𝜑) ⇒ ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | ||
Theorem | sb4e 1798 | One direction of a simplified definition of substitution that unlike sb4 1825 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑)) | ||
Theorem | hbsb2a 1799 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) | ||
Theorem | hbsb2e 1800 | Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.) |
⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |