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Theorem List for Metamath Proof Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdeqeq 3301 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 3302 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 3303* 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 3304* Variation of nfcdeq 3303 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝐴    &   CondEq(𝑥 = 𝑦𝐴 = 𝐵)       𝐴 = 𝐵

Theoremru 3305 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 4629 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 4617, Pairing prex 4735, Union uniex 6727, Power Set pwex 4673, and Infinity omex 8299 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 5775 (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 9056 and Cantor's Theorem canth 6385 are provably false! (See ncanth 6386 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 4607 replaces ax-rep 4597) with ax-sep 4607 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 8263 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (theorem ruv 8266). See ruALT 8267 for an alternate proof of ru 3305 derived from that fact. (Contributed by NM, 7-Aug-1994.)

{𝑥𝑥𝑥} ∉ V

2.1.9  Proper substitution of classes for sets

Syntaxwsbc 3306 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 3307 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 3333 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 3308 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 3308, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3307 in the form of sbc8g 3314. 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 3307 and assert that [𝐴 / 𝑥]𝜑 is always false when 𝐴 is a proper class.

The theorem sbc2or 3315 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3308.

The related definition df-csb 3404 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 3308 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 3307 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 3309 instead of df-sbc 3307. (dfsbcq2 3309 is needed because unlike Quine we do not overload the df-sb 1831 syntax.) As a consequence of these theorems, we can derive sbc8g 3314, which is a weaker version of df-sbc 3307 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 3314, so we will allow direct use of df-sbc 3307 after theorem sbc2or 3315 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 3309 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 1831 and substitution for class variables df-sbc 3307. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3308. (Contributed by NM, 31-Dec-2016.)
(𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))

Theoremsbsbc 3310 Show that df-sb 1831 and df-sbc 3307 are equivalent when the class term 𝐴 in df-sbc 3307 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1831 for proofs involving df-sbc 3307. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)
([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)

Theoremsbceq1d 3311 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 3312 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 3313* Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐵 / 𝑥]𝜒))

Theoremsbc8g 3314 This is the closest we can get to df-sbc 3307 if we start from dfsbcq 3308 (see its comments) and dfsbcq2 3309. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑}))

Theoremsbc2or 3315* 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 3331 (false) and sbc6 3333 (true) conclusions. This is interesting since dfsbcq 3308 and dfsbcq2 3309 (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 3316 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 3317 Equality theorem for class substitution. Class version of sbequ12 2130. (Contributed by NM, 26-Sep-2003.)
(𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))

Theoremsbceq2a 3318 Equality theorem for class substitution. Class version of sbequ12r 2131. (Contributed by NM, 4-Jan-2017.)
(𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑𝜑))

Theoremspsbc 3319 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 2245 and rspsbc 3388. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))

Theoremspsbcd 3320 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 2245 and rspsbc 3388. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑𝐴𝑉)    &   (𝜑 → ∀𝑥𝜓)       (𝜑[𝐴 / 𝑥]𝜓)

Theoremsbcth 3321 A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)
𝜑       (𝐴𝑉[𝐴 / 𝑥]𝜑)

Theoremsbcthdv 3322* Deduction version of sbcth 3321. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
(𝜑𝜓)       ((𝜑𝐴𝑉) → [𝐴 / 𝑥]𝜓)

Theoremsbcid 3323 An identity theorem for substitution. See sbid 2133. (Contributed by Mario Carneiro, 18-Feb-2017.)
([𝑥 / 𝑥]𝜑𝜑)

Theoremnfsbc1d 3324 Deduction version of nfsbc1 3325. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
(𝜑𝑥𝐴)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑥]𝜓)

Theoremnfsbc1 3325 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 3326* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
𝑥[𝐴 / 𝑥]𝜑

Theoremnfsbcd 3327 Deduction version of nfsbc 3328. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)

Theoremnfsbc 3328 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 3329* 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 3330* 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 3331* 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 3332* 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 3333* 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 3334* An equivalence for class substitution in the spirit of df-clab 2501. 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 3335 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 3336* 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 3337* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3338.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
((𝐴𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓))) → ([𝐴 / 𝑥]𝜑𝜓))

Theoremsbciegf 3338* 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 3339* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))

Theoremsbcie2g 3340* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3341 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 3341* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑𝜓)

Theoremsbciedf 3342* 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 3343* 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 3344* 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 3345 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3233 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 3346* Substitution applied to an atomic wff. Set theory version of eqsb3 2619. (Contributed by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵𝐴 = 𝐵))

Theoremsbcng 3347 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))

Theoremsbcimg 3348 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcan 3349 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 3350 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 3351 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
(𝐴𝑉 → ([𝐴 / 𝑥](𝜑𝜓) ↔ ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)))

Theoremsbcn1 3352 Move negation in and out of class substitution. One direction of sbcng 3347 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥] ¬ 𝜑 → ¬ [𝐴 / 𝑥]𝜑)

Theoremsbcim1 3353 Distribution of class substitution over implication. One direction of sbcimg 3348 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcbi1 3354 Distribution of class substitution over biconditional. One direction of sbcbig 3351 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥](𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcbi2 3355 Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
(∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Theoremsbcal 3356* 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 3357* 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 3358* Set theory version of sbeqal1 37517. (Contributed by Andrew Salmon, 28-Jun-2011.)
(𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))

Theoremsbeqalb 3359* 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 3360 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbcbidv 3361* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))

Theoremsbcbii 3362 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
(𝜑𝜓)       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓)

Theoremeqsbc3r 3363* eqsbc3 3346 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 3364* Obsolete proof of eqsbc3r 3363 as of 7-Jul-2021. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 37994 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 3365 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 3366* Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)
([𝐴 / 𝑥]𝑥𝐵𝐴𝐵)

Theoremsbcel2gv 3367* 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 3368* Class substitution into a membership relation. One direction of sbcel2gv 3367 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
([𝐵 / 𝑥]𝐴𝑥𝐴𝐵)

Theoremsbcimdv 3369* Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1714). (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 3370* Obsolete proof of sbcimdv 3369 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 3371 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 3372 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 3373 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 3374* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3372. (Contributed by Alan Sare, 10-Nov-2012.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜑))

Theoremsbc2iegf 3375* 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 3376* 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 3377* 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))

Theoremsbc3ie 3378* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))       ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑𝜓)

Theoremsbccomlem 3379* Lemma for sbccom 3380. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theoremsbccom 3380* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐵 / 𝑦][𝐴 / 𝑥]𝜑)

Theoremsbcralt 3381* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
((𝐴𝑉𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theoremsbcrext 3382* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)
(𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))

TheoremsbcrextOLD 3383* Obsolete proof of sbcrext 3382 as of 7-Jul-2021. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theoremsbcralg 3384* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥]𝜑))

Theoremsbcrex 3385* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)

Theoremsbcreu 3386* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.)
([𝐴 / 𝑥]∃!𝑦𝐵 𝜑 ↔ ∃!𝑦𝐵 [𝐴 / 𝑥]𝜑)

Theoremsbcabel 3387* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
𝑥𝐵       (𝐴𝑉 → ([𝐴 / 𝑥]{𝑦𝜑} ∈ 𝐵 ↔ {𝑦[𝐴 / 𝑥]𝜑} ∈ 𝐵))

Theoremrspsbc 3388* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 2245 and spsbc 3319. See also rspsbca 3389 and rspcsbela 3861. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(𝐴𝐵 → (∀𝑥𝐵 𝜑[𝐴 / 𝑥]𝜑))

Theoremrspsbca 3389* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
((𝐴𝐵 ∧ ∀𝑥𝐵 𝜑) → [𝐴 / 𝑥]𝜑)

Theoremrspesbca 3390* Existence form of rspsbca 3389. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
((𝐴𝐵[𝐴 / 𝑥]𝜑) → ∃𝑥𝐵 𝜑)

Theoremspesbc 3391 Existence form of spsbc 3319. (Contributed by Mario Carneiro, 18-Nov-2016.)
([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Theoremspesbcd 3392 form of spsbc 3319. (Contributed by Mario Carneiro, 9-Feb-2017.)
(𝜑[𝐴 / 𝑥]𝜓)       (𝜑 → ∃𝑥𝜓)

Theoremsbcth2 3393* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
(𝑥𝐵𝜑)       (𝐴𝐵[𝐴 / 𝑥]𝜑)

Theoremra4v 3394* Version of ra4 3395 with a dv condition, requiring fewer axioms. This is stdpc5v 1820 for a restricted domain. (Contributed by BJ, 27-Mar-2020.)
(∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))

Theoremra4 3395 Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is the axiom stdpc5 2039 of standard predicate calculus for a restricted domain. See ra4v 3394 for a version requiring fewer axioms. (Contributed by NM, 16-Jan-2004.) (Proof shortened by BJ, 27-Mar-2020.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) → (𝜑 → ∀𝑥𝐴 𝜓))

Theoremrmo2 3396* Alternate definition of restricted "at most one." Note that ∃*𝑥𝐴𝜑 is not equivalent to 𝑦𝐴𝑥𝐴(𝜑𝑥 = 𝑦) (in analogy to reu6 3266); to see this, let 𝐴 be the empty set. However, one direction of this pattern holds; see rmo2i 3397. (Contributed by NM, 17-Jun-2017.)
𝑦𝜑       (∃*𝑥𝐴 𝜑 ↔ ∃𝑦𝑥𝐴 (𝜑𝑥 = 𝑦))

Theoremrmo2i 3397* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
𝑦𝜑       (∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦) → ∃*𝑥𝐴 𝜑)

Theoremrmo3 3398* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
𝑦𝜑       (∃*𝑥𝐴 𝜑 ↔ ∀𝑥𝐴𝑦𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))

Theoremrmob 3399* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
(𝑥 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝐶 → (𝜑𝜒))       ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))

Theoremrmoi 3400* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
(𝑥 = 𝐵 → (𝜑𝜓))    &   (𝑥 = 𝐶 → (𝜑𝜒))       ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓) ∧ (𝐶𝐴𝜒)) → 𝐵 = 𝐶)

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