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

Statement | ||

Theorem | 2reu5lem3 3101* | Lemma for 2reu5 3102. 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 3206. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5 3102* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2338 and reu3 3084. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | nelrdva 3103* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |

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 3108 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 3113 and cdeqab 3111. | ||

Syntax | wcdeq 3104 | 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 3105 | 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 3106 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqri 3107 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqth 3108 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqnot 3109 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

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

CondEq CondEq | ||

Theorem | cdeqab 3111* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal1 3112* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab1 3113* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

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

CondEq CondEq CondEq | ||

Theorem | cdeqcv 3115 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqeq 3116 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqel 3117 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | nfcdeq 3118* | 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 3119* | Variation of nfcdeq 3118 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

2.1.8 Russell's Paradox | ||

Theorem | ru 3120 |
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 4307 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 4299, Pairing prex 4366, Union uniex 4664, Power Set pwex 4342, and Infinity omex 7554 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 5490 (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 8311 and Cantor's Theorem canth 6498 are provably false! (See
ncanth 6499 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 4290
replaces ax-rep 4280) with ax-sep 4290 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 7521 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7524). See ruALT 7525 for an alternate proof of ru 3120 derived from that fact. (Contributed by NM, 7-Aug-1994.) |

2.1.9 Proper substitution of classes for
sets | ||

Syntax | wsbc 3121 | 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 3122 |
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 3147 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 3123 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 3129 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3123. The related definition df-csb 3212 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 3123 |
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 3122 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 3124 instead of df-sbc 3122. (dfsbcq2 3124 is needed because
unlike Quine we do not overload the df-sb 1656 syntax.) As a consequence of
these theorems, we can derive sbc8g 3128, which is a weaker version of
df-sbc 3122 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3128, so we will allow direct use of df-sbc 3122 after theorem sbc2or 3129 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 3124 | 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 3122. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3123. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbsbc 3125 | Show that df-sb 1656 and df-sbc 3122 are equivalent when the class term in df-sbc 3122 is a set variable. This theorem lets us reuse theorems based on df-sb 1656 for proofs involving df-sbc 3122. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |

Theorem | sbceq1d 3126 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbceq1dd 3127 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbc8g 3128 | This is the closest we can get to df-sbc 3122 if we start from dfsbcq 3123 (see its comments) and dfsbcq2 3124. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |

Theorem | sbc2or 3129* | 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 3145 (false) and sbc6 3147 (true) conclusions. This is interesting since dfsbcq 3123 and dfsbcq2 3124 (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 3130 | 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 3131 | Equality theorem for class substitution. Class version of sbequ12 1940. (Contributed by NM, 26-Sep-2003.) |

Theorem | sbceq2a 3132 | Equality theorem for class substitution. Class version of sbequ12r 1941. (Contributed by NM, 4-Jan-2017.) |

Theorem | spsbc 3133 | 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 2073 and rspsbc 3199. (Contributed by NM, 16-Jan-2004.) |

Theorem | spsbcd 3134 | 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 2073 and rspsbc 3199. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbcth 3135 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |

Theorem | sbcthdv 3136* | Deduction version of sbcth 3135. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbcid 3137 | An identity theorem for substitution. See sbid 1943. (Contributed by Mario Carneiro, 18-Feb-2017.) |

Theorem | nfsbc1d 3138 | Deduction version of nfsbc1 3139. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1 3139 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1v 3140* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbcd 3141 | Deduction version of nfsbc 3142. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc 3142 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbcco 3143* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcco2 3144* | 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 3145* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbc6g 3146* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbc6 3147* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |

Theorem | sbc7 3148* | An equivalence for class substitution in the spirit of df-clab 2391. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | cbvsbc 3149 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | cbvsbcv 3150* | 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 3151* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3152.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbciegf 3152* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcieg 3153* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |

Theorem | sbcie2g 3154* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 3155 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |

Theorem | sbcie 3155* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |

Theorem | sbciedf 3156* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcied 3157* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |

Theorem | sbcied2 3158* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |

Theorem | elrabsf 3159 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3051 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 3160* | Substitution applied to an atomic wff. Set theory version of eqsb3 2505. (Contributed by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcng 3161 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcimg 3162 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcan 3163 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcang 3164 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |

Theorem | sbcor 3165 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcorg 3166 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |

Theorem | sbcbig 3167 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |

Theorem | sbcal 3168* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcalg 3169* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcex2 3170* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |

Theorem | sbcexg 3171* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |

Theorem | sbceqal 3172* | Set theory version of sbeqal1 27465. (Contributed by Andrew Salmon, 28-Jun-2011.) |

Theorem | sbeqalb 3173* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |

Theorem | sbcbid 3174 | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcbidv 3175* | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcbii 3176 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) |

Theorem | sbcbiiOLD 3177 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |

Theorem | eqsbc3r 3178* | eqsbc3 3160 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 28661 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) |

Theorem | sbc3ang 3179 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcel1gv 3180* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcel2gv 3181* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcimdv 3182* | Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.) |

Theorem | sbctt 3183 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |

Theorem | sbcgf 3184 | 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 3185 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |

Theorem | sbcg 3186* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3184. (Contributed by Alan Sare, 10-Nov-2012.) |

Theorem | sbc2iegf 3187* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |

Theorem | sbc2ie 3188* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |

Theorem | sbc2iedv 3189* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |

Theorem | sbc3ie 3190* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) |

Theorem | sbccomlem 3191* | Lemma for sbccom 3192. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) |

Theorem | sbccom 3192* | Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |

Theorem | sbcralt 3193* | Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.) |

Theorem | sbcrext 3194* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcralg 3195* | Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcrexg 3196* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcreug 3197* | Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) |

Theorem | sbcabel 3198* | Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.) |

Theorem | rspsbc 3199* | 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 2073 and spsbc 3133. See also rspsbca 3200 and rspcsbela 3268. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |

Theorem | rspsbca 3200* | Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.) |

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