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Theorem List for Intuitionistic Logic Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnf2 1601 An alternate definition of df-nf 1393, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Theoremnf3 1602 An alternate definition of df-nf 1393. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))

Theoremnf4dc 1603 Variable 𝑥 is effectively not free in 𝜑 iff 𝜑 is always true or always false, given a decidability condition. The reverse direction, nf4r 1604, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))

Theoremnf4r 1604 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 1603. (Contributed by Jim Kingdon, 21-Jul-2018.)
((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Theorem19.36i 1605 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
𝑥𝜓    &   𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorem19.36-1 1606 Closed form of 19.36i 1605. 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.)
𝑥𝜓       (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))

Theorem19.37-1 1607 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.)
𝑥𝜑       (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))

Theorem19.37aiv 1608* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.38 1609 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.23t 1610 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(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))

Theorem19.23 1611 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
𝑥𝜓       (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.32dc 1612 Theorem 19.32 of [Margaris] p. 90, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
𝑥𝜑       (DECID 𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))

Theorem19.32r 1613 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1612. (Contributed by Jim Kingdon, 28-Jul-2018.)
𝑥𝜑       ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.31r 1614 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.)
𝑥𝜓       ((∀𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.44 1615 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45 1616 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theorem19.34 1617 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.41h 1618 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1619 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.41 1619 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.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.42h 1620 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1621 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theorem19.42 1621 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theoremexcom13 1622 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)

Theoremexrot3 1623 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)

Theoremexrot4 1624 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝑥𝑦𝜑)

Theoremnexr 1625 Inference from 19.8a 1525. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ ∃𝑥𝜑        ¬ 𝜑

Theoremexan 1626 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)

Theoremhbexd 1627 Deduction form of bound-variable hypothesis builder hbex 1570. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∃𝑦𝜓 → ∀𝑥𝑦𝜓))

Theoremeeor 1628 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

1.3.8  Equality theorems without distinct variables

Theorema9e 1629 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 1379 through ax-14 1448 and ax-17 1462, all axioms other than ax-9 1467 are believed to be theorems of free logic, although the system without ax-9 1467 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
𝑥 𝑥 = 𝑦

Theorema9ev 1630* At least one individual exists. Weaker version of a9e 1629. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦

Theoremax9o 1631 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theoremequid 1632 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

Theoremnfequid 1633 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.)
𝑦 𝑥 = 𝑥

Theoremstdpc6 1634 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1697.) 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

Theoremequcomi 1635 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremax6evr 1636* A commuted form of a9ev 1630. 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

Theoremequcom 1637 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcoms 1638 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦𝜑)       (𝑦 = 𝑥𝜑)

Theoremequtr 1639 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))

Theoremequtrr 1640 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 = 𝑥𝑧 = 𝑦))

Theoremequtr2 1641 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)

Theoremequequ1 1642 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequequ2 1643 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremelequ1 1644 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremelequ2 1645 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax11i 1646 Inference that has ax-11 1440 (without 𝑦) as its conclusion and doesn't require ax-10 1439, ax-11 1440, or ax-12 1445 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 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

1.3.9  Axioms ax-10 and ax-11

Theoremax10o 1647 Show that ax-10o 1648 can be derived from ax-10 1439. An open problem is whether this theorem can be derived from ax-10 1439 and the others when ax-11 1440 is replaced with ax-11o 1748. See theorem ax10 1649 for the rederivation of ax-10 1439 from ax10o 1647.

Normally, ax10o 1647 should be used rather than ax-10o 1648, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Axiomax-10o 1648 Axiom ax-10o 1648 ("o" for "old") was the original version of ax-10 1439, before it was discovered (in May 2008) that the shorter ax-10 1439 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 1647.

Normally, ax10o 1647 should be used rather than ax-10o 1648, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremax10 1649 Rederivation of ax-10 1439 from original version ax-10o 1648. See theorem ax10o 1647 for the derivation of ax-10o 1648 from ax-10 1439.

This theorem should not be referenced in any proof. Instead, use ax-10 1439 above so that uses of ax-10 1439 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremhbae 1650 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 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremnfae 1651 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧𝑥 𝑥 = 𝑦

Theoremhbaes 1652 Rule that applies hbae 1650 to antecedent. (Contributed by NM, 5-Aug-1993.)
(∀𝑧𝑥 𝑥 = 𝑦𝜑)       (∀𝑥 𝑥 = 𝑦𝜑)

Theoremhbnae 1653 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)

Theoremnfnae 1654 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧 ¬ ∀𝑥 𝑥 = 𝑦

Theoremhbnaes 1655 Rule that applies hbnae 1653 to antecedent. (Contributed by NM, 5-Aug-1993.)
(∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑥 𝑥 = 𝑦𝜑)

Theoremnaecoms 1656 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑥 𝑥 = 𝑦𝜑)       (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Theoremequs4 1657 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremequsalh 1658 A useful equivalence related to substitution. New proofs should use equsal 1659 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsal 1659 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.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsex 1660 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(𝜓 → ∀𝑥𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexd 1661 Deduction form of equsex 1660. (Contributed by Jim Kingdon, 29-Dec-2017.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜒))

Theoremdral1 1662 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 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremdral2 1663 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 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremdrex2 1664 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 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝜑 ↔ ∃𝑧𝜓))

Theoremdrnf1 1665 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑦𝜓))

Theoremdrnf2 1666 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
(∀𝑥 𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜓))

Theoremspimth 1667 Closed theorem form of spim 1670. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)
(∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Theoremspimt 1668 Closed theorem form of spim 1670. (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 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Theoremspimh 1669 Specialization, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. The spim 1670 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 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspim 1670 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1670 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.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimeh 1671 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 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspimed 1672 Deduction version of spime 1673. (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 → (𝜑𝜓))       (𝜒 → (𝜑 → ∃𝑥𝜓))

Theoremspime 1673 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.)
𝑥𝜑    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremcbv3 1674 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbv3h 1675 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 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbv1 1676 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.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv1h 1677 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 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Theoremcbv2h 1678 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
(𝜑 → (𝜓 → ∀𝑦𝜓))    &   (𝜑 → (𝜒 → ∀𝑥𝜒))    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbv2 1679 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.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Theoremcbvalh 1680 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 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbval 1681 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexh 1682 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 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremcbvex 1683 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremchvar 1684 Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓

Theoremequvini 1685 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 = 𝑧𝑧 = 𝑦))

Theoremequveli 1686 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 1685.) (Contributed by NM, 1-Mar-2013.) (Revised by NM, 3-Feb-2015.)
(∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)

Theoremnfald 1687 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.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

Theoremnfexd 1688 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.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)

1.3.10  Substitution (without distinct variables)

Syntaxwsb 1689 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

Definitiondf-sb 1690 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 1702.

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 1765, sbcom2 1908 and sbid2v 1917).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 1701 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 1912 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 1915.

When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 1812 and sb6 1811.

In classical logic, another possible definition is (𝑥 = 𝑦𝜑) ∨ ∀𝑥(𝑥 = 𝑦𝜑) 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 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))

Theoremsbimi 1691 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)

Theoremsbbii 1692 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)

Theoremsb1 1693 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremsb2 1694 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
(∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)

Theoremsbequ1 1695 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑))

Theoremsbequ2 1696 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))

Theoremstdpc7 1697 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1634.) 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]𝜑𝜑))

Theoremsbequ12 1698 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))

Theoremsbequ12r 1699 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
(𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Theoremsbequ12a 1700 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))

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