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Theorem List for Metamath Proof Explorer - 3201-3300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsbcng 3201 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Theoremsbcimg 3202 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)

Theoremsbcan 3203 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)

Theoremsbcang 3204 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)

Theoremsbcor 3205 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)

Theoremsbcorg 3206 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)

Theoremsbcbig 3207 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremsbcal 3208* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)

Theoremsbcalg 3209* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Theoremsbcex2 3210* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Theoremsbcexg 3211* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Theoremsbceqal 3212* Set theory version of sbeqal1 27574. (Contributed by Andrew Salmon, 28-Jun-2011.)

Theoremsbeqalb 3213* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)

Theoremsbcbid 3214 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Theoremsbcbidv 3215* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Theoremsbcbii 3216 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)

TheoremsbcbiiOLD 3217 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)

Theoremeqsbc3r 3218* eqsbc3 3200 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 28952 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremsbc3ang 3219 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcel1gv 3220* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcel2gv 3221* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcimdv 3222* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)

Theoremsbctt 3223 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremsbcgf 3224 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.)

Theoremsbc19.21g 3225 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Theoremsbcg 3226* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3224. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremsbc2iegf 3227* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2ie 3228* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2iedv 3229* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbc3ie 3230* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremsbccomlem 3231* Lemma for sbccom 3232. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)

Theoremsbccom 3232* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbcralt 3233* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)

Theoremsbcrext 3234* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremsbcralg 3235* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcrexg 3236* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcreug 3237* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)

Theoremsbcabel 3238* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)

Theoremrspsbc 3239* 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 2091 and spsbc 3173. See also rspsbca 3240 and rspcsbela 3308. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremrspsbca 3240* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)

Theoremrspesbca 3241* Existence form of rspsbca 3240. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremspesbc 3242 Existence form of spsbc 3173. (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremspesbcd 3243 form of spsbc 3173. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbcth2 3244* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremra5 3245 Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is an axiom of a predicate calculus for a restricted domain. Compare the unrestricted stdpc5 1816. (Contributed by NM, 16-Jan-2004.)

Theoremrmo2 3246* Alternate definition of restricted "at most one." Note that is not equivalent to (in analogy to reu6 3123); to see this, let be the empty set. However, one direction of this pattern holds; see rmo2i 3247. (Contributed by NM, 17-Jun-2017.)

Theoremrmo2i 3247* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)

Theoremrmo3 3248* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Theoremrmob 3249* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)

Theoremrmoi 3250* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

2.1.10  Proper substitution of classes for sets into classes

Syntaxcsb 3251 Extend class notation to include the proper substitution of a class for a set into another class.

Definitiondf-csb 3252* Define the proper substitution of a class for a set into another class. The underlined brackets distinguish it from the substitution into a wff, wsbc 3161, to prevent ambiguity. Theorem sbcel1g 3270 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3279 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)

Theoremcsb2 3253* Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that can be free in but cannot occur in . (Contributed by NM, 2-Dec-2013.)

Theoremcsbeq1 3254 Analog of dfsbcq 3163 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcbvcsb 3255 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on . (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcbvcsbv 3256* Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbeq1d 3257 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)

Theoremcsbid 3258 Analog of sbid 1947 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbeq1a 3259 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbco 3260* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbexg 3261 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Theoremcsbex 3262 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremcsbtt 3263 Substitution doesn't affect a constant (in which is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremcsbconstgf 3264 Substitution doesn't affect a constant (in which is not free). (Contributed by NM, 10-Nov-2005.)

Theoremcsbconstg 3265* Substitution doesn't affect a constant (in which is not free). csbconstgf 3264 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)

Theoremsbcel12g 3266 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbceqg 3267 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcnel12g 3268 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)

Theoremsbcne12g 3269 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)

Theoremsbcel1g 3270* Move proper substitution in and out of a membership relation. Note that the scope of is the wff , whereas the scope of is the class . (Contributed by NM, 10-Nov-2005.)

Theoremsbceq1g 3271* Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremsbcel2g 3272* Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.)

Theoremsbceq2g 3273* Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.)

Theoremcsbcomg 3274* Commutative law for double substitution into a class. (Contributed by NM, 14-Nov-2005.)

Theoremcsbeq2d 3275 Formula-building deduction rule for class substitution. (Contributed by NM, 22-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2dv 3276* Formula-building deduction rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbeq2i 3277 Formula-building inference rule for class substitution. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 1-Sep-2015.)

Theoremcsbvarg 3278 The proper substitution of a class for set variable results in the class (if the class exists). (Contributed by NM, 10-Nov-2005.)

Theoremsbccsbg 3279* Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.)

Theoremsbccsb2g 3280 Substitution into a wff expressed in using substitution into a class. (Contributed by NM, 27-Nov-2005.)

Theoremnfcsb1d 3281 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1 3282 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb1v 3283* Bound-variable hypothesis builder for substitution into a class. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsbd 3284 Deduction version of nfcsb 3285. (Contributed by NM, 21-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfcsb 3285 Bound-variable hypothesis builder for substitution into a class. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremcsbhypf 3286* Introduce an explicit substitution into an implicit substitution hypothesis. See sbhypf 3001 for class substitution version. (Contributed by NM, 19-Dec-2008.)

Theoremcsbiebt 3287* Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3291.) (Contributed by NM, 11-Nov-2005.)

Theoremcsbiedf 3288* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)

Theoremcsbieb 3289* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008.)

Theoremcsbiebg 3290* Bidirectional conversion between an implicit class substitution hypothesis and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)

Theoremcsbiegf 3291* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbief 3292* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbied 3293* Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbied2 3294* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)

Theoremcsbie2t 3295* Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3296). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcsbie2 3296* Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Theoremcsbie2g 3297* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3195 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremsbcnestgf 3298 Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)

Theoremcsbnestgf 3299 Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Theoremsbcnestg 3300* Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

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