P. 31 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3001-3100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rmo3f 3001*	Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
Theorem	rmo4f 3002*	Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦))
Theorem	reueq 3003*	Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵)
Theorem	rmoan 3004	Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
Theorem	rmoim 3005	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))
Theorem	rmoimia 3006	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	rmoimi2 3007	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓))    ⇒   ⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	2reuswapdc 3008*	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
⊢ (DECID ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)))
Theorem	reuind 3009*	Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ ((∀𝑥∀𝑦(((𝐴 ∈ 𝐶 ∧ 𝜑) ∧ (𝐵 ∈ 𝐶 ∧ 𝜓)) → 𝐴 = 𝐵) ∧ ∃𝑥(𝐴 ∈ 𝐶 ∧ 𝜑)) → ∃!𝑧 ∈ 𝐶 ∀𝑥((𝐴 ∈ 𝐶 ∧ 𝜑) → 𝑧 = 𝐴))
Theorem	2rmorex 3010*	Double restricted quantification with "at most one," analogous to 2moex 2164. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	nelrdva 3011*	Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐴)
2.1.7  Conditional equality (experimental) This is a very useless definition, which "abbreviates" (𝑥 = 𝑦 → 𝜑) as CondEq(𝑥 = 𝑦 → 𝜑). What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific ternary operation (𝑥 = 𝑦 → 𝜑). This is all used as part of a metatheorem: we want to say that ⊢ (𝑥 = 𝑦 → (𝜑(𝑥) ↔ 𝜑(𝑦))) and ⊢ (𝑥 = 𝑦 → 𝐴(𝑥) = 𝐴(𝑦)) are provable, for any expressions 𝜑(𝑥) or 𝐴(𝑥) in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables condition: every variable in 𝜑(𝑥) is assumed disjoint from 𝑥 except 𝑥 itself. For such a proof by induction, we must consider each of the possible forms of 𝜑(𝑥). If it is a variable other than 𝑥, then we have CondEq(𝑥 = 𝑦 → 𝐴 = 𝐴) or CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜑)), which is provable by cdeqth 3016 and reflexivity. Since we are only working with class and wff expressions, it can't be 𝑥 itself in set.mm, but if it was we'd have to also prove CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦) (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to 𝑥 or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that 𝑦 is disjoint from 𝜑(𝑥) and vice versa, because this is maintained through the induction. This is how we satisfy the disjoint variable conditions of cdeqab1 3021 and cdeqab 3019.
Syntax	wcdeq 3012	Extend wff notation to include conditional equality. This is a technical device used in the proof that Ⅎ is the not-free predicate, and that definitions are conservative as a result.
wff CondEq(𝑥 = 𝑦 → 𝜑)
Definition	df-cdeq 3013	Define conditional equality. All the notation to the left of the ↔ is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq𝑥𝑦𝜑. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (CondEq(𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
Theorem	cdeqi 3014	Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ CondEq(𝑥 = 𝑦 → 𝜑)
Theorem	cdeqri 3015	Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (𝑥 = 𝑦 → 𝜑)
Theorem	cdeqth 3016	Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝜑    ⇒   ⊢ CondEq(𝑥 = 𝑦 → 𝜑)
Theorem	cdeqnot 3017	Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
Theorem	cdeqal 3018*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Theorem	cdeqab 3019*	Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → {𝑧 ∣ 𝜑} = {𝑧 ∣ 𝜓})
Theorem	cdeqal1 3020*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	cdeqab1 3021*	Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓})
Theorem	cdeqim 3022	Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ CondEq(𝑥 = 𝑦 → (𝜒 ↔ 𝜃))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	cdeqcv 3023	Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦)
Theorem	cdeqeq 3024	Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
Theorem	cdeqel 3025	Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ⊢ CondEq(𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
Theorem	nfcdeq 3026*	If we have a conditional equality proof, where 𝜑 is 𝜑(𝑥) and 𝜓 is 𝜑(𝑦), and 𝜑(𝑥) in fact does not have 𝑥 free in it according to Ⅎ, then 𝜑(𝑥) ↔ 𝜑(𝑦) unconditionally. This proves that Ⅎ𝑥𝜑 is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	nfccdeq 3027*	Variation of nfcdeq 3026 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
2.1.8  Russell's Paradox
Theorem	ru 3028	Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14. In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as 𝐴 ∈ V, asserted that any collection of sets 𝐴 is a set i.e. belongs to the universe V of all sets. In particular, by substituting {𝑥 ∣ 𝑥 ∉ 𝑥} (the "Russell class") for 𝐴, it asserted {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 4205. (Contributed by NM, 7-Aug-1994.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V
2.1.9  Proper substitution of classes for sets
Syntax	wsbc 3029	Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class 𝐴 for setvar variable 𝑥 in wff 𝜑".
wff [𝐴 / 𝑥]𝜑
Definition	df-sbc 3030	Define the proper substitution of a class for a set. When 𝐴 is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3055 for our definition, which always evaluates to true for proper classes. Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 3031 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of 𝜑, and it does not seem possible to express it with a single closed formula. If we did not want to commit to any specific proper class behavior, we could use this definition only to prove Theorem dfsbcq 3031, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3030 in the form of sbc8g 3037. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of 𝐴 in every use of this definition) we allow direct reference to df-sbc 3030 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. The related definition df-csb defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
Theorem	dfsbcq 3031	This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 3030 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 3032 instead of df-sbc 3030. (dfsbcq2 3032 is needed because unlike Quine we do not overload the df-sb 1809 syntax.) As a consequence of these theorems, we can derive sbc8g 3037, which is a weaker version of df-sbc 3030 that leaves substitution undefined when 𝐴 is a proper class. However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3037, so we will allow direct use of df-sbc 3030. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.)
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜑))
Theorem	dfsbcq2 3032	This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1809 and substitution for class variables df-sbc 3030. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3031. (Contributed by NM, 31-Dec-2016.)
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	sbsbc 3033	Show that df-sb 1809 and df-sbc 3030 are equivalent when the class term 𝐴 in df-sbc 3030 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1809 for proofs involving df-sbc 3030. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbceq1d 3034	Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓))
Theorem	sbceq1dd 3035	Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)    ⇒   ⊢ (𝜑 → [𝐵 / 𝑥]𝜓)
Theorem	sbceqbid 3036*	Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒))
Theorem	sbc8g 3037	This is the closest we can get to df-sbc 3030 if we start from dfsbcq 3031 (see its comments) and dfsbcq2 3032. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
Theorem	sbcex 3038	By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)
⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
Theorem	sbceq1a 3039	Equality theorem for class substitution. Class version of sbequ12 1817. (Contributed by NM, 26-Sep-2003.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	sbceq2a 3040	Equality theorem for class substitution. Class version of sbequ12r 1818. (Contributed by NM, 4-Jan-2017.)
⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	spsbc 3041	Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1821 and rspsbc 3113. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
Theorem	spsbcd 3042	Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1821 and rspsbc 3113. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
Theorem	sbcth 3043	A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)
⊢ 𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → [𝐴 / 𝑥]𝜑)
Theorem	sbcthdv 3044*	Deduction version of sbcth 3043. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓)
Theorem	sbcid 3045	An identity theorem for substitution. See sbid 1820. (Contributed by Mario Carneiro, 18-Feb-2017.)
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
Theorem	nfsbc1d 3046	Deduction version of nfsbc1 3047. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)
Theorem	nfsbc1 3047	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
Theorem	nfsbc1v 3048*	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
Theorem	nfsbcd 3049	Deduction version of nfsbc 3050. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Theorem	nfsbc 3050	Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑
Theorem	sbcco 3051*	A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	sbcco2 3052*	A composition law for class substitution. Importantly, 𝑥 may occur free in the class expression substituted for 𝐴. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	sbc5 3053*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	sbc6g 3054*	An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	sbc6 3055*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑))
Theorem	sbc7 3056*	An equivalence for class substitution in the spirit of df-clab 2216. Note that 𝑥 and 𝐴 don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 ∧ [𝑦 / 𝑥]𝜑))
Theorem	cbvsbcw 3057*	Version of cbvsbc 3058 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbc 3058	Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbcv 3059*	Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	sbciegft 3060*	Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3061.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbciegf 3061*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbcieg 3062*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbcie2g 3063*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 3064 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒))
Theorem	sbcie 3064*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
Theorem	sbciedf 3065*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbcied 3066*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbcied2 3067*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	elrabsf 3068	Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2958 has implicit substitution). The hypothesis specifies that 𝑥 must not be a free variable in 𝐵. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))
Theorem	eqsbc1 3069*	Substitution for the left-hand side in an equality. Class version of eqsb1 2333. (Contributed by Andrew Salmon, 29-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
Theorem	sbcng 3070	Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
Theorem	sbcimg 3071	Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
Theorem	sbcan 3072	Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))
Theorem	sbcang 3073	Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓)))
Theorem	sbcor 3074	Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))
Theorem	sbcorg 3075	Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓)))
Theorem	sbcbig 3076	Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 ↔ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)))
Theorem	sbcn1 3077	Move negation in and out of class substitution. One direction of sbcng 3070 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)
Theorem	sbcim1 3078	Distribution of class substitution over implication. One direction of sbcimg 3071 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
Theorem	sbcbi1 3079	Distribution of class substitution over biconditional. One direction of sbcbig 3076 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))
Theorem	sbcbi2 3080	Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))
Theorem	sbcal 3081*	Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
Theorem	sbcalg 3082*	Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
Theorem	sbcex2 3083*	Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
Theorem	sbcexg 3084*	Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
Theorem	sbceqal 3085*	A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
Theorem	sbeqalb 3086*	Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
⊢ (𝐴 ∈ 𝑉 → ((∀𝑥(𝜑 ↔ 𝑥 = 𝐴) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝐵)) → 𝐴 = 𝐵))
Theorem	sbcbid 3087	Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))
Theorem	sbcbidv 3088*	Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))
Theorem	sbcbii 3089	Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)
Theorem	eqsbc2 3090*	Substitution for the right-hand side in an equality. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴))
Theorem	sbc3an 3091	Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓 ∧ [𝐴 / 𝑥]𝜒))
Theorem	sbcel1v 3092*	Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)
Theorem	sbcel2gv 3093*	Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (𝐵 ∈ 𝑉 → ([𝐵 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵))
Theorem	sbcel21v 3094*	Class substitution into a membership relation. One direction of sbcel2gv 3093 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)
Theorem	sbcimdv 3095*	Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1503). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
Theorem	sbctt 3096	Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbcgf 3097	Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbc19.21g 3098	Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝐴 / 𝑥]𝜓)))
Theorem	sbcg 3099*	Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3097. (Contributed by Alan Sare, 10-Nov-2012.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbc2iegf 3100*	Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ Ⅎ𝑥 𝐵 ∈ 𝑊    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))