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	cdeqi 3701	Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ CondEq(𝑥 = 𝑦 → 𝜑)
Theorem	cdeqri 3702	Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (𝑥 = 𝑦 → 𝜑)
Theorem	cdeqth 3703	Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝜑    ⇒   ⊢ CondEq(𝑥 = 𝑦 → 𝜑)
Theorem	cdeqnot 3704	Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))
Theorem	cdeqal 3705*	Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))
Theorem	cdeqab 3706*	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 3707*	Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 2373. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))
Theorem	cdeqab1 3708*	Distribute conditional equality over abstraction. Usage of this theorem is discouraged because it depends on ax-13 2373. (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 3709	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 3710	Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ CondEq(𝑥 = 𝑦 → 𝑥 = 𝑦)
Theorem	cdeqeq 3711	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 3712	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 3713*	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 2373. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝜑    &   ⊢ CondEq(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	nfccdeq 3714*	Variation of nfcdeq 3713 for classes. Usage of this theorem is discouraged because it depends on ax-13 2373. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-11 2155. (Revised by Gino Giotto, 19-May-2023.) (New usage is discouraged.)
⊢ Ⅎ𝑥𝐴    &   ⊢ CondEq(𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ 𝐴 = 𝐵
2.1.8  Russell's Paradox
Theorem	rru 3715*	Relative version of Russell's paradox ru 3716 (which corresponds to the case 𝐴 = V). Originally a subproof in pwnss 5273. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3051. (Revised by Steven Nguyen, 23-Nov-2022.) Reduce axiom usage. (Revised by Gino Giotto, 30-Aug-2024.)
⊢ ¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴
Theorem	ru 3716	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 5246 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 5232, Pairing prex 5356, Union uniex 7603, Power Set pwex 5304, and Infinity omex 9410 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 6529 (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 10240 and Cantor's theorem canth 7238 are provably false. (See ncanth 7239 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 5224 replaces ax-rep 5210) with ax-sep 5224 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 9364 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9370). See ruALT 9371 for an alternate proof of ru 3716 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2373. (Revised by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V
2.1.9  Proper substitution of classes for sets
Syntax	wsbc 3717	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 3718	Define the proper substitution of a class for a set. When 𝐴 is a proper class, our definition evaluates to false (see sbcex 3727). This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3749 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 3719 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 3719, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3718 in the form of sbc8g 3725. 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 3718 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class. Theorem sbc2or 3726 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3719. The related definition df-csb 3834 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 3719	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 3718 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 3720 instead of df-sbc 3718. (dfsbcq2 3720 is needed because unlike Quine we do not overload the df-sb 2069 syntax.) As a consequence of these theorems, we can derive sbc8g 3725, which is a weaker version of df-sbc 3718 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 3725, so we will allow direct use of df-sbc 3718 after Theorem sbc2or 3726 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 3720	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 2069 and substitution for class variables df-sbc 3718. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3719. (Contributed by NM, 31-Dec-2016.)
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	sbsbc 3721	Show that df-sb 2069 and df-sbc 3718 are equivalent when the class term 𝐴 in df-sbc 3718 is a setvar variable. This theorem lets us reuse theorems based on df-sb 2069 for proofs involving df-sbc 3718. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
Theorem	sbceq1d 3722	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 3723	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 3724*	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 3725	This is the closest we can get to df-sbc 3718 if we start from dfsbcq 3719 (see its comments) and dfsbcq2 3720. (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 3726*	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 3745 (false) and sbc6 3749 (true) conclusions. This is interesting since dfsbcq 3719 and dfsbcq2 3720 (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 3727	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 3728	Equality theorem for class substitution. Class version of sbequ12 2245. (Contributed by NM, 26-Sep-2003.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
Theorem	sbceq2a 3729	Equality theorem for class substitution. Class version of sbequ12r 2246. (Contributed by NM, 4-Jan-2017.)
⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	spsbc 3730	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 2072 and rspsbc 3813. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑))
Theorem	spsbcd 3731	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 2072 and rspsbc 3813. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)
Theorem	sbcth 3732	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 3733*	Deduction version of sbcth 3732. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [𝐴 / 𝑥]𝜓)
Theorem	sbcid 3734	An identity theorem for substitution. See sbid 2249. (Contributed by Mario Carneiro, 18-Feb-2017.)
⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)
Theorem	nfsbc1d 3735	Deduction version of nfsbc1 3736. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)
Theorem	nfsbc1 3736	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 3737*	Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
Theorem	nfsbcdw 3738*	Deduction version of nfsbcw 3739. Version of nfsbcd 3741 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 23-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
Theorem	nfsbcw 3739*	Bound-variable hypothesis builder for class substitution. Version of nfsbc 3742 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 7-Sep-2014.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑
Theorem	sbccow 3740*	A composition law for class substitution. Version of sbcco 3743 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 26-Sep-2003.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
Theorem	nfsbcd 3741	Deduction version of nfsbc 3742. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker nfsbcdw 3738 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 3742	Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker nfsbcw 3739 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 3743*	A composition law for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker sbccow 3740 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 3744*	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 3745*	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 3746*	Alternate proof of sbc5 3745. This proof helps show how clelab 2884 works, since it is equivalent but shorter thanks to now-available library theorems like vtoclbg 3508 and isset 3446. (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 3747*	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	sbc6gOLD 3748*	Obsolete version of sbc6g 3747 as of 5-Oct-2024. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
Theorem	sbc6 3749*	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 3750*	An equivalence for class substitution in the spirit of df-clab 2717. 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 3751*	Change bound variables in a wff substitution. Version of cbvsbc 3753 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by Jeff Hankins, 19-Sep-2009.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbcvw 3752*	Change the bound variable of a class substitution using implicit substitution. Version of cbvsbcv 3754 with a disjoint variable condition, which does not require ax-13 2373. (Contributed by NM, 30-Sep-2008.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓)
Theorem	cbvsbc 3753	Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker cbvsbcw 3751 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 3754*	Change the bound variable of a class substitution using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker cbvsbcvw 3752 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 3755*	Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3756.) (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 3756*	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 3757*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2138, ax-12 2172. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbciegOLD 3758*	Obsolete version of sbcieg 3757 as of 12-Oct-2024. (Contributed by NM, 10-Nov-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
Theorem	sbcie2g 3759*	Conversion of implicit substitution to explicit class substitution. This version of sbcie 3760 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 3760*	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 3761*	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 3762*	Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2138, ax-12 2172. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbciedOLD 3763*	Obsolete version of sbcied 3762 as of 12-Oct-2024. (Contributed by NM, 13-Dec-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
Theorem	sbcied2 3764*	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 3765	Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3621 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 3766*	Substitution for the left-hand side in an equality. Class version of eqsb1 2866. (Contributed by Andrew Salmon, 29-Jun-2011.) Avoid ax-13 2373. (Revised by Wolf Lammen, 29-Apr-2023.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
Theorem	sbcng 3767	Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
Theorem	sbcimg 3768	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 3769	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 3770	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 3771	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 3772	Move negation in and out of class substitution. One direction of sbcng 3767 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)
Theorem	sbcim1 3773	Distribution of class substitution over implication. One direction of sbcimg 3768 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 26-Oct-2024.)
⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
Theorem	sbcim1OLD 3774	Obsolete version of sbcim1 3773 as of 26-Oct-2024. (Contributed by NM, 17-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓))
Theorem	sbcbid 3775	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 3776*	Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) Drop ax-12 2172. (Revised by Gino Giotto, 1-Dec-2023.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜒))
Theorem	sbcbii 3777	Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓)
Theorem	sbcbi1 3778	Distribution of class substitution over biconditional. One direction of sbcbig 3771 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐴 / 𝑥](𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))
Theorem	sbcbi2 3779	Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) (Proof shortened by Wolf Lammen, 4-May-2023.) Avoid ax-10, ax-12. (Revised by Steven Nguyen, 5-May-2024.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))
Theorem	sbcbi2OLD 3780	Obsolete proof of sbcbi2 3779 as of 5-May-2024. (Contributed by Giovanni Mascellani, 9-Apr-2018.) (Proof shortened by Wolf Lammen, 4-May-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜓))
Theorem	sbcal 3781*	Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)
Theorem	sbcex2 3782*	Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.)
⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)
Theorem	sbceqal 3783*	Class version of one implication of equvelv 2035. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by SN, 26-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
Theorem	sbceqalOLD 3784*	Obsolete version of sbceqal 3783 as of 26-Oct-2024. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝑥 = 𝐵) → 𝐴 = 𝐵))
Theorem	sbeqalb 3785*	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	eqsbc2 3786*	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 3787	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 3788*	Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) Avoid ax-13 2373. (Revised by Wolf Lammen, 30-Apr-2023.)
⊢ ([𝐴 / 𝑥]𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)
Theorem	sbcel2gv 3789*	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 3790*	Class substitution into a membership relation. One direction of sbcel2gv 3789 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
⊢ ([𝐵 / 𝑥]𝐴 ∈ 𝑥 → 𝐴 ∈ 𝐵)
Theorem	sbcimdv 3791*	Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1813). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
Theorem	sbcimdvOLD 3792*	Obsolete version of sbcimdv 3791 as of 12-Oct-2024. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
Theorem	sbctt 3793	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 3794	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 3795	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 3796*	Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3794. (Contributed by Alan Sare, 10-Nov-2012.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbcgOLD 3797*	Obsolete version of sbcg 3796 as of 12-Oct-2024. (Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
Theorem	sbcgfi 3798	Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
⊢ 𝐴 ∈ V    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)
Theorem	sbc2iegf 3799*	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))
Theorem	sbc2ie 3800*	Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Gino Giotto, 12-Oct-2024.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)