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Theorem List for Metamath Proof Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrmoim 3401 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))

Theoremrmoimia 3402 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝑥𝐴 → (𝜑𝜓))       (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑)

Theoremrmoimi2 3403 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜓))       (∃*𝑥𝐵 𝜓 → ∃*𝑥𝐴 𝜑)

Theorem2reuswap 3404* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))

Theoremreuind 3405* Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = 𝑦𝐴 = 𝐵)       ((∀𝑥𝑦(((𝐴𝐶𝜑) ∧ (𝐵𝐶𝜓)) → 𝐴 = 𝐵) ∧ ∃𝑥(𝐴𝐶𝜑)) → ∃!𝑧𝐶𝑥((𝐴𝐶𝜑) → 𝑧 = 𝐴))

Theorem2rmorex 3406* Double restricted quantification with "at most one," analogous to 2moex 2541. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)

Theorem2reu5lem1 3407* Lemma for 2reu5 3410. Note that ∃!𝑥𝐴∃!𝑦𝐵𝜑 does not mean "there is exactly one 𝑥 in 𝐴 and exactly one 𝑦 in 𝐵 such that 𝜑 holds;" see comment for 2eu5 2555. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥∃!𝑦(𝑥𝐴𝑦𝐵𝜑))

Theorem2reu5lem2 3408* Lemma for 2reu5 3410. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥∃*𝑦(𝑥𝐴𝑦𝐵𝜑))

Theorem2reu5lem3 3409* Lemma for 2reu5 3410. 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 3519. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∀𝑥𝐴 ∃*𝑦𝐵 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝑤𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2reu5 3410* Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2555 and reu3 3390. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∀𝑥𝐴 ∃*𝑦𝐵 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theoremnelrdva 3411* Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
((𝜑𝑥𝐴) → 𝑥𝐵)       (𝜑 → ¬ 𝐵𝐴)

2.1.7  Conditional equality (experimental)

This is a very useless definition, which "abbreviates" (𝑥 = 𝑦𝜑) as CondEq(𝑥 = 𝑦𝜑). 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 3-ary operation (𝑥 = 𝑦𝜑).

This is all used as part of a metatheorem: we want to say that (𝑥 = 𝑦 → (𝜑(𝑥) ↔ 𝜑(𝑦))) and (𝑥 = 𝑦𝐴(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 = 𝑦𝐴 = 𝐴) or CondEq(𝑥 = 𝑦 → (𝜑𝜑)), which is provable by cdeqth 3416 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 = 𝑦𝑥 = 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 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 dv conditions of cdeqab1 3421 and cdeqab 3419.

Syntaxwcdeq 3412 Extend wff notation to include conditional equality. This is a technical device used in the proof that is the not-free predicate, and that definitions are conservative as a result.
wff CondEq(𝑥 = 𝑦𝜑)

Definitiondf-cdeq 3413 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 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜑))

Theoremcdeqi 3414 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝑥 = 𝑦𝜑)       CondEq(𝑥 = 𝑦𝜑)

Theoremcdeqri 3415 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝜑)       (𝑥 = 𝑦𝜑)

Theoremcdeqth 3416 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝜑       CondEq(𝑥 = 𝑦𝜑)

Theoremcdeqnot 3417 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (¬ 𝜑 ↔ ¬ 𝜓))

Theoremcdeqal 3418* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (∀𝑧𝜑 ↔ ∀𝑧𝜓))

Theoremcdeqab 3419* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑧𝜑} = {𝑧𝜓})

Theoremcdeqal1 3420* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremcdeqab1 3421* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))       CondEq(𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜓})

Theoremcdeqim 3422 Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦 → (𝜑𝜓))    &   CondEq(𝑥 = 𝑦 → (𝜒𝜃))       CondEq(𝑥 = 𝑦 → ((𝜑𝜒) ↔ (𝜓𝜃)))

Theoremcdeqcv 3423 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝑥 = 𝑦)

Theoremcdeqeq 3424 Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝐴 = 𝐵)    &   CondEq(𝑥 = 𝑦𝐶 = 𝐷)       CondEq(𝑥 = 𝑦 → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremcdeqel 3425 Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq(𝑥 = 𝑦𝐴 = 𝐵)    &   CondEq(𝑥 = 𝑦𝐶 = 𝐷)       CondEq(𝑥 = 𝑦 → (𝐴𝐶𝐵𝐷))

Theoremnfcdeq 3426* 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 , then 𝜑(𝑥) ↔ 𝜑(𝑦) unconditionally. This proves that 𝑥𝜑 is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   CondEq(𝑥 = 𝑦 → (𝜑𝜓))       (𝜑𝜓)

Theoremnfccdeq 3427* Variation of nfcdeq 3426 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   CondEq(𝑥 = 𝑦𝐴 = 𝐵)       𝐴 = 𝐵

Theoremru 3428 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 {𝑥𝑥𝑥} (the "Russell class") for 𝐴, it asserted {𝑥𝑥𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥𝑥𝑥} ∉ 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 ssex 4793 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 4781, Pairing prex 4900, Union uniex 6938, Power Set pwex 4839, and Infinity omex 8525 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 5964 (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, contradicting ZF and NBG set theories, and it has other bizarre consequences: when sets become too huge (beyond the size of those used in standard mathematics), the Axiom of Choice ac4 9282 and Cantor's Theorem canth 6593 are provably false! (See ncanth 6594 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 4772 replaces ax-rep 4762) with ax-sep 4772 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 8489 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv 8492). See ruALT 8493 for an alternate proof of ru 3428 derived from that fact. (Contributed by NM, 7-Aug-1994.)

{𝑥𝑥𝑥} ∉ V

2.1.9  Proper substitution of classes for sets

Syntaxwsbc 3429 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

Definitiondf-sbc 3430 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 3456 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 3431 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 3431, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3430 in the form of sbc8g 3437. 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 3430 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3438 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3431.

The related definition df-csb 3527 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]𝜑𝐴 ∈ {𝑥𝜑})

Theoremdfsbcq 3431 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 3430 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 3432 instead of df-sbc 3430. (dfsbcq2 3432 is needed because unlike Quine we do not overload the df-sb 1879 syntax.) As a consequence of these theorems, we can derive sbc8g 3437, which is a weaker version of df-sbc 3430 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 3437, so we will allow direct use of df-sbc 3430 after theorem sbc2or 3438 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))

Theoremdfsbcq2 3432 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 1879 and substitution for class variables df-sbc 3430. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3431. (Contributed by NM, 31-Dec-2016.)
(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))

Theoremsbsbc 3433 Show that df-sb 1879 and df-sbc 3430 are equivalent when the class term 𝐴 in df-sbc 3430 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1879 for proofs involving df-sbc 3430. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)

Theoremsbceq1d 3434 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
(𝜑𝐴 = 𝐵)       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜓))

Theoremsbceq1dd 3435 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)
(𝜑𝐴 = 𝐵)    &   (𝜑[𝐴 / 𝑥]𝜓)       (𝜑[𝐵 / 𝑥]𝜓)

Theoremsbceqbid 3436* Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))

Theoremsbc8g 3437 This is the closest we can get to df-sbc 3430 if we start from dfsbcq 3431 (see its comments) and dfsbcq2 3432. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))

Theoremsbc2or 3438* 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 3454 (false) and sbc6 3456 (true) conclusions. This is interesting since dfsbcq 3431 and dfsbcq2 3432 (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 = 𝐴𝜑)) ∨ ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))

Theoremsbcex 3439 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]𝜑𝐴 ∈ V)

Theoremsbceq1a 3440 Equality theorem for class substitution. Class version of sbequ12 2109. (Contributed by NM, 26-Sep-2003.)
(𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))

Theoremsbceq2a 3441 Equality theorem for class substitution. Class version of sbequ12r 2110. (Contributed by NM, 4-Jan-2017.)
(𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))

Theoremspsbc 3442 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 2351 and rspsbc 3511. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))

Theoremspsbcd 3443 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 2351 and rspsbc 3511. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝐴𝑉)    &   (𝜑 → ∀𝑥𝜓)       (𝜑[𝐴 / 𝑥]𝜓)

Theoremsbcth 3444 A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)
𝜑       (𝐴𝑉[𝐴 / 𝑥]𝜑)

Theoremsbcthdv 3445* Deduction version of sbcth 3444. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝜑𝜓)       ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)

Theoremsbcid 3446 An identity theorem for substitution. See sbid 2112. (Contributed by Mario Carneiro, 18-Feb-2017.)
([𝑥 / 𝑥]𝜑𝜑)

Theoremnfsbc1d 3447 Deduction version of nfsbc1 3448. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

Theoremnfsbc1 3448 Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴       𝑥[𝐴 / 𝑥]𝜑

Theoremnfsbc1v 3449* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥[𝐴 / 𝑥]𝜑

Theoremnfsbcd 3450 Deduction version of nfsbc 3451. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Theoremnfsbc 3451 Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑥𝐴    &   𝑥𝜑       𝑥[𝐴 / 𝑦]𝜑

Theoremsbcco 3452* 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)

Theoremsbcco2 3453* 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 = 𝑦𝐴 = 𝐵)       ([𝑥 / 𝑦][𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)

Theoremsbc5 3454* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))

Theoremsbc6g 3455* An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴𝜑)))

Theoremsbc6 3456* 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 = 𝐴𝜑))

Theoremsbc7 3457* An equivalence for class substitution in the spirit of df-clab 2607. 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))

Theoremcbvsbc 3458 Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)

Theoremcbvsbcv 3459* 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 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)

Theoremsbciegft 3460* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3461.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))

Theoremsbciegf 3461* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
𝑥𝜓    &   (𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))

Theoremsbcieg 3462* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))

Theoremsbcie2g 3463* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3464 avoids a disjointness condition on 𝑥, 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑦 = 𝐴 → (𝜓𝜒))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜒))

Theoremsbcie 3464* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑𝜓)

Theoremsbciedf 3465* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))    &   𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))

Theoremsbcied 3466* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))

Theoremsbcied2 3467* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐴 = 𝐵)    &   ((𝜑𝑥 = 𝐵) → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓𝜒))

Theoremelrabsf 3468 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3354 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.)
𝑥𝐵       (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵[𝐴 / 𝑥]𝜑))

Theoremeqsbc3 3469* Substitution applied to an atomic wff. Set theory version of eqsb3 2726. (Contributed by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))

Theoremsbcng 3470 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))

Theoremsbcimg 3471 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcan 3472 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcor 3473 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcbig 3474 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcn1 3475 Move negation in and out of class substitution. One direction of sbcng 3470 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)

Theoremsbcim1 3476 Distribution of class substitution over implication. One direction of sbcimg 3471 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcbi1 3477 Distribution of class substitution over biconditional. One direction of sbcbig 3474 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcbi2 3478 Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
(∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcal 3479* 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)

Theoremsbcex2 3480* 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)

Theoremsbceqal 3481* Set theory version of sbeqal1 38418. (Contributed by Andrew Salmon, 28-Jun-2011.)
(𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))

Theoremsbeqalb 3482* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
(𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → 𝐴 = 𝐵))

Theoremsbcbid 3483 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbcbidv 3484* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbcbii 3485 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
(𝜑𝜓)       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Theoremeqsbc3r 3486* eqsbc3 3469 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))

Theoremeqsbc3rOLD 3487* Obsolete proof of eqsbc3r 3486 as of 7-Jul-2021. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 38895 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥𝐵 = 𝐴))

Theoremsbc3an 3488 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓𝜒) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbcel1v 3489* Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)

Theoremsbcel2gv 3490* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐵𝑉 → ([𝐵 / 𝑥]𝐴𝑥𝐴𝐵))

Theoremsbcel21v 3491* Class substitution into a membership relation. One direction of sbcel2gv 3490 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐵 / 𝑥]𝐴𝑥𝐴𝐵)

Theoremsbcimdv 3492* Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1736). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

TheoremsbcimdvOLD 3493* Obsolete proof of sbcimdv 3492 as of 7-Jul-2021. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbctt 3494 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))

Theoremsbcgf 3495 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.)
𝑥𝜑       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))

Theoremsbc19.21g 3496 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
𝑥𝜑       (𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ (𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcg 3497* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3495. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))

Theoremsbc2iegf 3498* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
𝑥𝜓    &   𝑦𝜓    &   𝑥 𝐵𝑊    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))

Theoremsbc2ie 3499* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)

Theoremsbc2iedv 3500* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))       (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓𝜒))

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392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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