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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | moeq3 3201* | "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.) |
Theorem | mosub 3202* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
Theorem | mo2icl 3203* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |
Theorem | mob2 3204* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |
Theorem | moi2 3205* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |
Theorem | mob 3206* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Theorem | moi 3207* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Theorem | morex 3208* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | euxfr2 3209* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |
Theorem | euxfr 3210* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |
Theorem | euind 3211* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
Theorem | reu2 3212* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
Theorem | reu6 3213* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
Theorem | reu3 3214* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu6i 3215* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | eqreu 3216* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | rmo4 3217* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
Theorem | reu4 3218* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
Theorem | reu7 3219* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu8 3220* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu2eqd 3221* | Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
Theorem | reueq 3222* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
Theorem | rmoan 3223 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
Theorem | rmoim 3224 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimia 3225 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimi2 3226 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reuswap 3227* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
Theorem | reuind 3228* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
Theorem | 2rmorex 3229* | Double restricted quantification with "at most one," analogous to 2moex 2296. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reu5lem1 3230* | Lemma for 2reu5 3233. Note that does not mean "there is exactly one in and exactly one in such that holds;" see comment for 2eu5 2313. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reu5lem2 3231* | Lemma for 2reu5 3233. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reu5lem3 3232* | Lemma for 2reu5 3233. 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 3341. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reu5 3233* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2313 and reu3 3214. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | nelrdva 3234* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |
This is a very useless definition, which "abbreviates" as CondEq . 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 . This is all used as part of a metatheorem: we want to say that and are provable, for any expressions or 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 is assumed disjoint from except itself. For such a proof by induction, we must consider each of the possible forms of . If it is a variable other than , then we have CondEq or CondEq , which is provable by cdeqth 3239 and reflexivity. Since we are only working with class and wff expressions, it can't be itself in set.mm, but if it was we'd have to also prove CondEq (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 or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that is disjoint from and vice versa, because this is maintained through the induction. This is how we satisfy the DV assumptions of cdeqab1 3244 and cdeqab 3242. | ||
Syntax | wcdeq 3235 | 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. |
CondEq | ||
Definition | df-cdeq 3236 | 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 CondEq. 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 | ||
Theorem | cdeqi 3237 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqri 3238 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqth 3239 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqnot 3240 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal 3241* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab 3242* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal1 3243* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab1 3244* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqim 3245 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqcv 3246 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqeq 3247 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqel 3248 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | nfcdeq 3249* | If we have a conditional equality proof, where is and is , and in fact does not have free in it according to , then unconditionally. This proves that is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | nfccdeq 3250* | Variation of nfcdeq 3249 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | ru 3251 |
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 , asserted that any collection of sets is a set i.e. belongs to the universe of all sets. In particular, by substituting (the "Russell class") for , it asserted , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove . 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 4509 asserting that is a set only when it is smaller than some other set . However, Zermelo was then faced with a "chicken and egg" problem of how to show is a set, leading him to introduce the set-building axioms of Null Set 0ex 4497, Pairing prex 4604, Union uniex 6495, Power Set pwex 4548, and Infinity omex 7974 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 5574 (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 8768 and Cantor's Theorem canth 6155 are provably false! (See ncanth 6156 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 4488 replaces ax-rep 4478) with ax-sep 4488 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 7938 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7941). See ruALT 7942 for an alternate proof of ru 3251 derived from that fact. (Contributed by NM, 7-Aug-1994.) |
Syntax | wsbc 3252 | Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class for setvar variable in wff ." |
Definition | df-sbc 3253 |
Define the proper substitution of a class for a set.
When is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 3279 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 3254 below). For example, if is a proper class, Quine's substitution of for in evaluates to rather than our falsehood. (This can be seen by substituting , , and for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) Unfortunately, Quine's definition requires a recursive syntactical breakdown of , 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 3254, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 3253 in the form of sbc8g 3260. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of in every use of this definition) we allow direct reference to df-sbc 3253 and assert that is always false when is a proper class. The theorem sbc2or 3261 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3254. The related definition df-csb 3349 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.) |
Theorem | dfsbcq 3254 |
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 3253 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 3255 instead of df-sbc 3253. (dfsbcq2 3255 is needed because unlike Quine we do not overload the df-sb 1748 syntax.) As a consequence of these theorems, we can derive sbc8g 3260, which is a weaker version of df-sbc 3253 that leaves substitution undefined when is a proper class. However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3260, so we will allow direct use of df-sbc 3253 after theorem sbc2or 3261 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.) |
Theorem | dfsbcq2 3255 | 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 1748 and substitution for class variables df-sbc 3253. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3254. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbsbc 3256 | Show that df-sb 1748 and df-sbc 3253 are equivalent when the class term in df-sbc 3253 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1748 for proofs involving df-sbc 3253. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
Theorem | sbceq1d 3257 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbceq1dd 3258 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbceqbid 3259* | Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
Theorem | sbc8g 3260 | This is the closest we can get to df-sbc 3253 if we start from dfsbcq 3254 (see its comments) and dfsbcq2 3255. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Theorem | sbc2or 3261* | 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 behavior at proper classes, matching the sbc5 3277 (false) and sbc6 3279 (true) conclusions. This is interesting since dfsbcq 3254 and dfsbcq2 3255 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem doesn't 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 that or may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.) |
Theorem | sbcex 3262 | 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.) |
Theorem | sbceq1a 3263 | Equality theorem for class substitution. Class version of sbequ12 2000. (Contributed by NM, 26-Sep-2003.) |
Theorem | sbceq2a 3264 | Equality theorem for class substitution. Class version of sbequ12r 2001. (Contributed by NM, 4-Jan-2017.) |
Theorem | spsbc 3265 | 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 2098 and rspsbc 3331. (Contributed by NM, 16-Jan-2004.) |
Theorem | spsbcd 3266 | 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 2098 and rspsbc 3331. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | sbcth 3267 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |
Theorem | sbcthdv 3268* | Deduction version of sbcth 3267. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbcid 3269 | An identity theorem for substitution. See sbid 2003. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Theorem | nfsbc1d 3270 | Deduction version of nfsbc1 3271. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1 3271 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1v 3272* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbcd 3273 | Deduction version of nfsbc 3274. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc 3274 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbcco 3275* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcco2 3276* | A composition law for class substitution. Importantly, may occur free in the class expression substituted for . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbc5 3277* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbc6g 3278* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbc6 3279* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
Theorem | sbc7 3280* | An equivalence for class substitution in the spirit of df-clab 2368. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | cbvsbc 3281 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | cbvsbcv 3282* | Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbciegft 3283* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3284.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbciegf 3284* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcieg 3285* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
Theorem | sbcie2g 3286* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 3287 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | sbcie 3287* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
Theorem | sbciedf 3288* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcied 3289* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | sbcied2 3290* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | elrabsf 3291 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3180 has implicit substitution). The hypothesis specifies that must not be a free variable in . (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Theorem | eqsbc3 3292* | Substitution applied to an atomic wff. Set theory version of eqsb3 2502. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcng 3293 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcimg 3294 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcan 3295 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.) |
Theorem | sbcor 3296 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 17-Aug-2018.) |
Theorem | sbcbig 3297 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
Theorem | sbcn1 3298 | Move negation in and out of class substitution. One direction of sbcng 3293 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcim1 3299 | Distribution of class substitution over implication. One direction of sbcimg 3294 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi1 3300 | Distribution of class substitution over biconditional. One direction of sbcbig 3297 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
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