Home Intuitionistic Logic ExplorerTheorem List (p. 30 of 135) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 2901-3000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdeqnot 2901 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqal 2902* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqab 2903* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqal1 2904* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqab1 2905* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqim 2906 Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremcdeqcv 2907 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqeq 2908 Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremcdeqel 2909 Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremnfcdeq 2910* 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

Theoremnfccdeq 2911* Variation of nfcdeq 2910 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremru 2912 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 asserting that is a set only when it is smaller than some other set . The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 4054. (Contributed by NM, 7-Aug-1994.)

2.1.9  Proper substitution of classes for sets

Syntaxwsbc 2913 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 ."

Definitiondf-sbc 2914 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 2938 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 2915 below). 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 2915, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2914 in the form of sbc8g 2920. 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 2914 and assert that is always false when is a proper class.

The related definition df-csb 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.)

Theoremdfsbcq 2915 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 2914 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 2916 instead of df-sbc 2914. (dfsbcq2 2916 is needed because unlike Quine we do not overload the df-sb 1737 syntax.) As a consequence of these theorems, we can derive sbc8g 2920, which is a weaker version of df-sbc 2914 that leaves substitution undefined when is a proper class.

However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2920, so we will allow direct use of df-sbc 2914. 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.)

Theoremdfsbcq2 2916 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 1737 and substitution for class variables df-sbc 2914. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2915. (Contributed by NM, 31-Dec-2016.)

Theoremsbsbc 2917 Show that df-sb 1737 and df-sbc 2914 are equivalent when the class term in df-sbc 2914 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1737 for proofs involving df-sbc 2914. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)

Theoremsbceq1d 2918 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)

Theoremsbceq1dd 2919 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)

Theoremsbc8g 2920 This is the closest we can get to df-sbc 2914 if we start from dfsbcq 2915 (see its comments) and dfsbcq2 2916. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)

Theoremsbcex 2921 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.)

Theoremsbceq1a 2922 Equality theorem for class substitution. Class version of sbequ12 1745. (Contributed by NM, 26-Sep-2003.)

Theoremsbceq2a 2923 Equality theorem for class substitution. Class version of sbequ12r 1746. (Contributed by NM, 4-Jan-2017.)

Theoremspsbc 2924 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 1749 and rspsbc 2995. (Contributed by NM, 16-Jan-2004.)

Theoremspsbcd 2925 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 1749 and rspsbc 2995. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbcth 2926 A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.)

Theoremsbcthdv 2927* Deduction version of sbcth 2926. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremsbcid 2928 An identity theorem for substitution. See sbid 1748. (Contributed by Mario Carneiro, 18-Feb-2017.)

Theoremnfsbc1d 2929 Deduction version of nfsbc1 2930. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbc1 2930 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbc1v 2931* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbcd 2932 Deduction version of nfsbc 2933. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremnfsbc 2933 Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremsbcco 2934* A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremsbcco2 2935* 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.)

Theoremsbc5 2936* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremsbc6g 2937* An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremsbc6 2938* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)

Theoremsbc7 2939* An equivalence for class substitution in the spirit of df-clab 2127. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremcbvsbcw 2940* Version of cbvsbc 2941 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)

Theoremcbvsbc 2941 Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremcbvsbcv 2942* Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremsbciegft 2943* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2944.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremsbciegf 2944* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Theoremsbcieg 2945* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)

Theoremsbcie2g 2946* Conversion of implicit substitution to explicit class substitution. This version of sbcie 2947 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremsbcie 2947* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)

Theoremsbciedf 2948* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)

Theoremsbcied 2949* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Theoremsbcied2 2950* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Theoremelrabsf 2951 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2842 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.)

Theoremeqsbc3 2952* Substitution applied to an atomic wff. Set theory version of eqsb3 2244. (Contributed by Andrew Salmon, 29-Jun-2011.)

Theoremsbcng 2953 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Theoremsbcimg 2954 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)

Theoremsbcan 2955 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)

Theoremsbcang 2956 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)

Theoremsbcor 2957 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)

Theoremsbcorg 2958 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)

Theoremsbcbig 2959 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremsbcn1 2960 Move negation in and out of class substitution. One direction of sbcng 2953 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Theoremsbcim1 2961 Distribution of class substitution over implication. One direction of sbcimg 2954 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Theoremsbcbi1 2962 Distribution of class substitution over biconditional. One direction of sbcbig 2959 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Theoremsbcbi2 2963 Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)

Theoremsbcal 2964* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)

Theoremsbcalg 2965* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Theoremsbcex2 2966* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Theoremsbcexg 2967* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)

Theoremsbceqal 2968* A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.)

Theoremsbeqalb 2969* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)

Theoremsbcbid 2970 Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)

Theoremsbcbidv 2971* Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.)

Theoremsbcbii 2972 Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.)

Theoremeqsbc3r 2973* eqsbc3 2952 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)

Theoremsbc3an 2974 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.)

Theoremsbcel1v 2975* Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.)

Theoremsbcel2gv 2976* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcel21v 2977* Class substitution into a membership relation. One direction of sbcel2gv 2976 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Theoremsbcimdv 2978* Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1434). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)

Theoremsbctt 2979 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)

Theoremsbcgf 2980 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 2981 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Theoremsbcg 2982* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2980. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremsbc2iegf 2983* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2ie 2984* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Theoremsbc2iedv 2985* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbc3ie 2986* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)

Theoremsbccomlem 2987* Lemma for sbccom 2988. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)

Theoremsbccom 2988* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Theoremsbcralt 2989* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)

Theoremsbcrext 2990* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremsbcralg 2991* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsbcrex 2992* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Theoremsbcreug 2993* Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.)

Theoremsbcabel 2994* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)

Theoremrspsbc 2995* 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 1749 and spsbc 2924. See also rspsbca 2996 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremrspsbca 2996* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)

Theoremrspesbca 2997* Existence form of rspsbca 2996. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Theoremspesbc 2998 Existence form of spsbc 2924. (Contributed by Mario Carneiro, 18-Nov-2016.)

Theoremspesbcd 2999 form of spsbc 2924. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbcth2 3000* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13441
 Copyright terms: Public domain < Previous  Next >