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Type | Label | Description |
---|---|---|

Statement | ||

2.1.7 Conditional equality
(experimental)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 3105 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 Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each set 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 3110 and cdeqab 3108. | ||

Syntax | wcdeq 3101 | 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 3102 | 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 3103 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqri 3104 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqth 3105 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqnot 3106 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal 3107* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab 3108* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal1 3109* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab1 3110* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqim 3111 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqcv 3112 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqeq 3113 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqel 3114 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | nfcdeq 3115* | 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 3116* | Variation of nfcdeq 3115 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

2.1.8 Russell's Paradox | ||

Theorem | ru 3117 |
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 4302 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 4294, Pairing prex 4361, Union uniex 4659, Power Set pwex 4337, and Infinity omex 7545 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 5485 (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 set 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 8302 and Cantor's Theorem canth 6489 are provably false! (See
ncanth 6490 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 4285
replaces ax-rep 4275) with ax-sep 4285 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," Under our ZF set theory, every set is a member of the Russell class by elirrv 7512 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7515). See ruALT 7516 for an alternate proof of ru 3117 derived from that fact. (Contributed by NM, 7-Aug-1994.) |

2.1.9 Proper substitution of classes for
sets | ||

Syntax | wsbc 3118 | Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class for set variable in wff ." |

Definition | df-sbc 3119 |
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 3144 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 3120 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 The theorem sbc2or 3126 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3120. The related definition df-csb 3209 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 3120 |
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 3119 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 3121 instead of df-sbc 3119. (dfsbcq2 3121 is needed because
unlike Quine we do not overload the df-sb 1656 syntax.) As a consequence of
these theorems, we can derive sbc8g 3125, which is a weaker version of
df-sbc 3119 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3125, so we will allow direct use of df-sbc 3119 after theorem sbc2or 3126 below. Proper substiution 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 3121 | 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 1656 and substitution for class variables df-sbc 3119. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3120. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbsbc 3122 | Show that df-sb 1656 and df-sbc 3119 are equivalent when the class term in df-sbc 3119 is a set variable. This theorem lets us reuse theorems based on df-sb 1656 for proofs involving df-sbc 3119. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |

Theorem | sbceq1d 3123 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbceq1dd 3124 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbc8g 3125 | This is the closest we can get to df-sbc 3119 if we start from dfsbcq 3120 (see its comments) and dfsbcq2 3121. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |

Theorem | sbc2or 3126* | 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 3142 (false) and sbc6 3144 (true) conclusions. This is interesting since dfsbcq 3120 and dfsbcq2 3121 (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 3127 | 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 3128 | Equality theorem for class substitution. Class version of sbequ12 1940. (Contributed by NM, 26-Sep-2003.) |

Theorem | sbceq2a 3129 | Equality theorem for class substitution. Class version of sbequ12r 1941. (Contributed by NM, 4-Jan-2017.) |

Theorem | spsbc 3130 | 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 2071 and rspsbc 3196. (Contributed by NM, 16-Jan-2004.) |

Theorem | spsbcd 3131 | 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 2071 and rspsbc 3196. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbcth 3132 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |

Theorem | sbcthdv 3133* | Deduction version of sbcth 3132. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbcid 3134 | An identity theorem for substitution. See sbid 1943. (Contributed by Mario Carneiro, 18-Feb-2017.) |

Theorem | nfsbc1d 3135 | Deduction version of nfsbc1 3136. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1 3136 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1v 3137* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbcd 3138 | Deduction version of nfsbc 3139. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc 3139 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbcco 3140* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcco2 3141* | 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 3142* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbc6g 3143* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbc6 3144* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |

Theorem | sbc7 3145* | An equivalence for class substitution in the spirit of df-clab 2388. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | cbvsbc 3146 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | cbvsbcv 3147* | 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 3148* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3149.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbciegf 3149* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcieg 3150* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |

Theorem | sbcie2g 3151* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 3152 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |

Theorem | sbcie 3152* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |

Theorem | sbciedf 3153* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcied 3154* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |

Theorem | sbcied2 3155* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |

Theorem | elrabsf 3156 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3048 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 3157* | Substitution applied to an atomic wff. Set theory version of eqsb3 2502. (Contributed by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcng 3158 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcimg 3159 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcan 3160 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcang 3161 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |

Theorem | sbcor 3162 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcorg 3163 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |

Theorem | sbcbig 3164 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |

Theorem | sbcal 3165* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcalg 3166* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcex2 3167* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |

Theorem | sbcexg 3168* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |

Theorem | sbceqal 3169* | Set theory version of sbeqal1 27282. (Contributed by Andrew Salmon, 28-Jun-2011.) |

Theorem | sbeqalb 3170* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |

Theorem | sbcbid 3171 | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcbidv 3172* | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcbii 3173 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) |

Theorem | sbcbiiOLD 3174 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | eqsbc3r 3175* | eqsbc3 3157 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 28309 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) |

Theorem | sbc3ang 3176 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcel1gv 3177* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcel2gv 3178* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcimdv 3179* | Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.) |

Theorem | sbctt 3180 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |

Theorem | sbcgf 3181 | 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.) |

Theorem | sbc19.21g 3182 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |

Theorem | sbcg 3183* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3181. (Contributed by Alan Sare, 10-Nov-2012.) |

Theorem | sbc2iegf 3184* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |

Theorem | sbc2ie 3185* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |

Theorem | sbc2iedv 3186* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |

Theorem | sbc3ie 3187* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) |

Theorem | sbccomlem 3188* | Lemma for sbccom 3189. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) |

Theorem | sbccom 3189* | Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |

Theorem | sbcralt 3190* | Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.) |

Theorem | sbcrext 3191* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcralg 3192* | Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcrexg 3193* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcreug 3194* | Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) |

Theorem | sbcabel 3195* | Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.) |

Theorem | rspsbc 3196* | 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 2071 and spsbc 3130. See also rspsbca 3197 and rspcsbela 3265. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |

Theorem | rspsbca 3197* | Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.) |

Theorem | rspesbca 3198* | Existence form of rspsbca 3197. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |

Theorem | spesbc 3199 | Existence form of spsbc 3130. (Contributed by Mario Carneiro, 18-Nov-2016.) |

Theorem | spesbcd 3200 | form of spsbc 3130. (Contributed by Mario Carneiro, 9-Feb-2017.) |

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