P. 38 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 3701-3800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rmo4f 3701*	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	reu2eqd 3702*	Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.)
⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → 𝐵 = 𝐶)
Theorem	reueq 3703*	Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)
⊢ (𝐵 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 𝑥 = 𝐵)
Theorem	rmoeq 3704*	Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by NM, 29-Oct-2020.)
⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴
Theorem	rmoan 3705	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 3706	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 3707	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	rmoimi 3708	Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	rmoimi2 3709	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	2reu5a 3710	Double restricted existential uniqueness in terms of restricted existence and restricted "at most one". (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑) ∧ ∃*𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∃*𝑦 ∈ 𝐵 𝜑)))
Theorem	reuimrmo 3711	Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2646. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))
Theorem	2reuswap 3712*	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
Theorem	2reuswap2 3713*	A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.)
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
Theorem	reuxfrd 3714*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) Separate variables 𝐵 and 𝐶. (Revised by Thierry Arnoux, 8-Oct-2017.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦 ∈ 𝐶 𝜓))
Theorem	reuxfr 3715*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴)    ⇒   ⊢ (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜑) ↔ ∃!𝑦 ∈ 𝐶 𝜑)
Theorem	reuxfr1d 3716*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr1ds 3717. (Contributed by Thierry Arnoux, 7-Apr-2017.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))
Theorem	reuxfr1ds 3717*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 5381 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))
Theorem	reuxfr1 3718*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 5382 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)
⊢ (𝑦 ∈ 𝐶 → 𝐴 ∈ 𝐵)    &   ⊢ (𝑥 ∈ 𝐵 → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐶 𝜓)
Theorem	reuind 3719*	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 3720*	Double restricted quantification with "at most one", analogous to 2moex 2670. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑)
Theorem	2reu5lem1 3721*	Lemma for 2reu5 3724. Note that ∃!𝑥 ∈ 𝐴∃!𝑦 ∈ 𝐵𝜑 does not mean "there is exactly one 𝑥 in 𝐴 and exactly one 𝑦 in 𝐵 such that 𝜑 holds"; see comment for 2eu5 2685. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥∃!𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
Theorem	2reu5lem2 3722*	Lemma for 2reu5 3724. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑))
Theorem	2reu5lem3 3723*	Lemma for 2reu5 3724. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3843. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	2reu5 3724*	Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2685 and reu3 3693. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ ((∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))
Theorem	2reurmo 3725	Double restricted quantification with restricted existential uniqueness and restricted "at most one", analogous to 2eumo 2672. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
Theorem	2reurex 3726*	Double restricted quantification with existential uniqueness, analogous to 2euex 2671. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
⊢ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝜑)
Theorem	2rmoswap 3727*	A condition allowing to swap restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2674. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))
Theorem	2rexreu 3728*	Double restricted existential uniqueness implies double restricted unique existential quantification, analogous to 2exeu 2676. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)
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 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 assumption: 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 3733 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 setvar 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 universal quantifier 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 3738 and cdeqab 3736.
Syntax	wcdeq 3729	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 3730	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 3731	Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ CondEq(𝑥 = 𝑦 → 𝜑)
Theorem	cdeqri 3732	Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (𝑥 = 𝑦 → 𝜑)
Theorem	cdeqth 3733	Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝜑    ⇒   ⊢ CondEq(𝑥 = 𝑦 → 𝜑)
Theorem	cdeqnot 3734	Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
Theorem	cdeqal 3735*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Theorem	cdeqab 3736*	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 3737*	Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	cdeqab1 3738*	Distribute conditional equality over abstraction. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓})
Theorem	cdeqim 3739	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 3740	Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦)
Theorem	cdeqeq 3741	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 3742	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 3743*	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. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	nfccdeq 3744*	Variation of nfcdeq 3743 for classes. Usage of this theorem is discouraged because it depends on ax-13 2406. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2194. (Revised by GG, 19-May-2023.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
2.1.8  Russell's Paradox
Theorem	rru 3745*	Relative version of Russell's paradox ru 3746 (which corresponds to the case 𝐴 = V). Originally a subproof in pwnss 5313. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3065. (Revised by Steven Nguyen, 23-Nov-2022.) Reduce axiom usage. (Revised by GG, 30-Aug-2024.)
⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴
Theorem	ru 3746	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, which Frege acknowledged in the second edition of his Grundgesetze der Arithmetik. In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 5282 asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. However, Zermelo was then faced with a "chicken and egg" problem of how to show 𝐵 is a set, leading him to introduce the set-building axioms of Null Set 0ex 5262, Pairing prex 5400, Union uniex 7728, Power Set pwex 5342, and Infinity omex 9600 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 6613 (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287). Russell himself continued in a different direction, avoiding the paradox with his "theory of types". Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contrarily to ZF and NBG set theories. Russell's paradox has other consequences: when classes are too large (beyond the size of those used in standard mathematics), the axiom of choice ac4 10447 and Cantor's theorem canth 7354 are provably false. (See ncanth 7355 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 5251 replaces ax-rep 5232) with ax-sep 5251 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic", J. Symb. Logic 9:1-19 (1944). Under our ZF set theory, every set is a member of the Russell class by elirrv 9547 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9558). See ruALT 9559 for an alternate proof of ru 3746 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2406. (Revised by BJ, 12-Oct-2019.) Remove use of ax-10 2178, ax-11 2194, and ax-12 2215. (Revised by BTernaryTau, 20-Jun-2025.) (Proof modification is discouraged.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V
2.1.9  Proper substitution of classes for sets
Syntax	wsbc 3747	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 3748	Define the proper substitution of a class for a set. When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3757). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3778 for our definition, whose right-hand side 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 3749 below). For example, if 𝐴 is a proper class, Quine's substitution of 𝐴 for 𝑦 in 0 ∈ 𝑦 evaluates to 0 ∈ 𝐴 rather than our falsehood. (This can be seen by substituting 𝐴, 𝑦, and 0 for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactic 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 3749, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3748 in the form of sbc8g 3755. 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 3748 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. Theorem sbc2or 3756 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3749. The related definition df-csb 3856 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 3749	Proper substitution of a class for a set in a wff given equal classes. This is the essence of the sixth axiom of Frege, specifically Proposition 52 of [Frege1879] p. 50. 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 3748 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 3750 instead of df-sbc 3748. (dfsbcq2 3750 is needed because unlike Quine we do not overload the df-sb 2094 syntax.) As a consequence of these theorems, we can derive sbc8g 3755, which is a weaker version of df-sbc 3748 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 3755, so we will allow direct use of df-sbc 3748 after Theorem sbc2or 3756 below. Proper substitution 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 3750	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 2094 and substitution for class variables df-sbc 3748. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3749. (Contributed by NM, 31-Dec-2016.)
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	sbsbc 3751	Show that df-sb 2094 and df-sbc 3748 are equivalent when the class term 𝐴 in df-sbc 3748 is a setvar variable. This theorem lets us reuse theorems based on df-sb 2094 for proofs involving df-sbc 3748. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbceq1d 3752	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 3753	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 3754*	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 3755	This is the closest we can get to df-sbc 3748 if we start from dfsbcq 3749 (see its comments) and dfsbcq2 3750. (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	sbc2or 3756*	The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for [𝐴 / 𝑥]𝜑 behavior at proper classes, matching the sbc5 3775 (false) and sbc6 3778 (true) conclusions. This is interesting since dfsbcq 3749 and dfsbcq2 3750 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem does not tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable 𝑦 that 𝜑 or 𝐴 may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.)
⊢ (([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	sbcex 3757	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 3758	Equality theorem for class substitution. Class version of sbequ12 2289. (Contributed by NM, 26-Sep-2003.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	sbceq2a 3759	Equality theorem for class substitution. Class version of sbequ12r 2290. (Contributed by NM, 4-Jan-2017.)
⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	spsbc 3760	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. This is Frege's ninth axiom per Proposition 58 of [Frege1879] p. 51. See also stdpc4 2101 and rspsbc 3835. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
Theorem	spsbcd 3761	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 2101 and rspsbc 3835. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
Theorem	sbcth 3762	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 3763*	Deduction version of sbcth 3762. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓)
Theorem	sbcid 3764	An identity theorem for substitution. See sbid 2293. (Contributed by Mario Carneiro, 18-Feb-2017.)
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
Theorem	nfsbc1d 3765	Deduction version of nfsbc1 3766. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)
Theorem	nfsbc1 3766	Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
Theorem	nfsbc1v 3767*	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
Theorem	nfsbcdw 3768*	Deduction version of nfsbcw 3769. Version of nfsbcd 3771 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 23-Nov-2005.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Theorem	nfsbcw 3769*	Bound-variable hypothesis builder for class substitution. Version of nfsbc 3772 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 7-Sep-2014.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑
Theorem	sbccow 3770*	A composition law for class substitution. Version of sbcco 3773 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 26-Sep-2003.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.)
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	nfsbcd 3771	Deduction version of nfsbc 3772. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker nfsbcdw 3768 when possible. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Theorem	nfsbc 3772	Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker nfsbcw 3769 when possible. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑
Theorem	sbcco 3773*	A composition law for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker sbccow 3770 when possible. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.)
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	sbcco2 3774*	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 3775*	An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by SN, 2-Sep-2024.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	sbc5ALT 3776*	Alternate proof of sbc5 3775. This proof helps show how clelab 2909 works, since it is equivalent but shorter thanks to now-available library theorems like vtoclbg 3527 and isset 3471. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	sbc6g 3777*	An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by SN, 5-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	sbc6 3778*	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 3779*	An equivalence for class substitution in the spirit of df-clab 2744. 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 3780*	Change bound variables in a wff substitution. Version of cbvsbc 3782 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by Jeff Hankins, 19-Sep-2009.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbcvw 3781*	Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3783 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 30-Sep-2008.) Avoid ax-13 2406. (Revised by GG, 10-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbc 3782	Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker cbvsbcw 3780 when possible. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbcv 3783*	Change the bound variable of a class substitution using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker cbvsbcvw 3781 when possible. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) (New usage is discouraged.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	sbciegft 3784*	Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3785.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) (Proof shortened by SN, 14-May-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbciegf 3785*	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 3786*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2178, ax-12 2215. (Revised by GG, 12-Oct-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbcie2g 3787*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 3788 avoids a disjointness condition on 𝑥, 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 3788*	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 3789*	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 3790*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2178, ax-12 2215. (Revised by GG, 12-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbcied2 3791*	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 3792	Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3650 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 3793*	Substitution for the left-hand side in an equality. Class version of eqsb1 2891. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2406. (Revised by Wolf Lammen, 29-Apr-2023.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
Theorem	sbcng 3794	Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
Theorem	sbcimg 3795	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 3796	Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∧ [𝐴 / 𝑥]𝜓))
Theorem	sbcor 3797	Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝐴 / 𝑥]𝜑 ∨ [𝐴 / 𝑥]𝜓))
Theorem	sbcbig 3798	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 3799	Move negation in and out of class substitution. One direction of sbcng 3794 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)
Theorem	sbcim1 3800	Distribution of class substitution over implication. One direction of sbcimg 3795 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2178, ax-12 2215. (Revised by SN, 26-Oct-2024.)
⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))