HomeHome Metamath Proof Explorer
Theorem List (p. 31 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21494)
  Hilbert Space Explorer  Hilbert Space Explorer
(21495-23017)
  Users' Mathboxes  Users' Mathboxes
(23018-31433)
 

Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbcex 3001 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.)
 |-  ( [. A  /  x ]. ph  ->  A  e.  _V )
 
Theoremsbceq1a 3002 Equality theorem for class substitution. Class version of sbequ12 1862. (Contributed by NM, 26-Sep-2003.)
 |-  ( x  =  A  ->  ( ph  <->  [. A  /  x ].
 ph ) )
 
Theoremsbceq2a 3003 Equality theorem for class substitution. Class version of sbequ12r 1863. (Contributed by NM, 4-Jan-2017.)
 |-  ( A  =  x 
 ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremspsbc 3004 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 1969 and rspsbc 3070. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( A. x ph  -> 
 [. A  /  x ].
 ph ) )
 
Theoremspsbcd 3005 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 1969 and rspsbc 3070. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  [. A  /  x ]. ps )
 
Theoremsbcth 3006 A substitution into a theorem remains true (when  A is a set). (Contributed by NM, 5-Nov-2005.)
 |-  ph   =>    |-  ( A  e.  V  -> 
 [. A  /  x ].
 ph )
 
Theoremsbcthdv 3007* Deduction version of sbcth 3006. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
 
Theoremsbcid 3008 An identity theorem for substitution. See sbid 1865. (Contributed by Mario Carneiro, 18-Feb-2017.)
 |-  ( [. x  /  x ]. ph  <->  ph )
 
Theoremnfsbc1d 3009 Deduction version of nfsbc1 3010. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x [. A  /  x ]. ps )
 
Theoremnfsbc1 3010 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbc1v 3011* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbcd 3012 Deduction version of nfsbc 3013. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x [. A  /  y ]. ps )
 
Theoremnfsbc 3013 Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x [. A  /  y ]. ph
 
Theoremsbcco 3014* A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ].
 ph )
 
Theoremsbcco2 3015* A composition law for class substitution. Importantly,  x may occur free in the class expression substituted for  A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( x  =  y 
 ->  A  =  B )   =>    |-  ( [. x  /  y ]. [. B  /  x ].
 ph 
 <-> 
 [. A  /  x ].
 ph )
 
Theoremsbc5 3016* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph )
 )
 
Theoremsbc6g 3017* An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
 ) )
 
Theoremsbc6 3018* An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
 )
 
Theoremsbc7 3019* An equivalence for class substitution in the spirit of df-clab 2271. Note that  x and  A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( [. A  /  x ]. ph  <->  E. y ( y  =  A  /\  [. y  /  x ]. ph )
 )
 
Theoremcbvsbc 3020 Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  y ]. ps )
 
Theoremcbvsbcv 3021* Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  y ]. ps )
 
Theoremsbciegft 3022* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3023.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/ x ps  /\ 
 A. x ( x  =  A  ->  ( ph 
 <->  ps ) ) ) 
 ->  ( [. A  /  x ]. ph  <->  ps ) )
 
Theoremsbciegf 3023* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |- 
 F/ x ps   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
 
Theoremsbcieg 3024* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)
 |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
 
Theoremsbcie2g 3025* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3026 avoids a disjointness condition on  x ,  A by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  (
 y  =  A  ->  ( ps  <->  ch ) )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ch ) )
 
Theoremsbcie 3026* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.)
 |-  A  e.  _V   &    |-  ( x  =  A  ->  (
 ph 
 <->  ps ) )   =>    |-  ( [. A  /  x ]. ph  <->  ps )
 
Theoremsbciedf 3027* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   &    |-  F/ x ph   &    |-  ( ph  ->  F/ x ch )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <->  ch ) )
 
Theoremsbcied 3028* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  =  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <->  ch ) )
 
Theoremsbcied2 3029* Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A  =  B )   &    |-  (
 ( ph  /\  x  =  B )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <->  ch ) )
 
Theoremelrabsf 3030 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2923 has implicit substitution). The hypothesis specifies that  x must not be a free variable in  B. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x B   =>    |-  ( A  e.  { x  e.  B  |  ph
 } 
 <->  ( A  e.  B  /\  [. A  /  x ].
 ph ) )
 
Theoremeqsbc3 3031* Substitution applied to an atomic wff. Set theory version of eqsb3 2385. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
 
Theoremsbcng 3032 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ].
 ph ) )
 
Theoremsbcimg 3033 Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps )  <->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps ) ) )
 
Theoremsbcan 3034 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 )
 
Theoremsbcang 3035 Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 ) )
 
Theoremsbcor 3036 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
 
Theoremsbcorg 3037 Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) ) )
 
Theoremsbcbig 3038 Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  <->  ps )  <->  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps ) ) )
 
Theoremsbcal 3039* Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph )
 
Theoremsbcalg 3040* Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  y ]. A. x ph  <->  A. x [. A  /  y ]. ph ) )
 
Theoremsbcex2 3041* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
 |-  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph )
 
Theoremsbcexg 3042* Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  y ]. E. x ph  <->  E. x [. A  /  y ]. ph ) )
 
Theoremsbceqal 3043* Set theory version of sbeqal1 27008. (Contributed by Andrew Salmon, 28-Jun-2011.)
 |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B )
 )
 
Theoremsbeqalb 3044* Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
 |-  ( A  e.  V  ->  ( ( A. x ( ph  <->  x  =  A )  /\  A. x (
 ph 
 <->  x  =  B ) )  ->  A  =  B ) )
 
Theoremsbcbid 3045 Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. A  /  x ].
 ch ) )
 
Theoremsbcbidv 3046* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. A  /  x ].
 ch ) )
 
Theoremsbcbii 3047 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
 |-  ( ph  <->  ps )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps )
 
TheoremsbcbiiOLD 3048 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.)
 |-  ( ph  <->  ps )   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps ) )
 
Theoremeqsbc3r 3049* eqsbc3 3031 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 27896 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A )
 )
 
Theoremsbc3ang 3050 Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  /\  ps  /\ 
 ch )  <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps  /\  [. A  /  x ]. ch ) ) )
 
Theoremsbcel1gv 3051* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  e.  B  <->  A  e.  B ) )
 
Theoremsbcel2gv 3052* Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( B  e.  V  ->  ( [. B  /  x ]. A  e.  x  <->  A  e.  B ) )
 
Theoremsbcimdv 3053* Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\  A  e.  V ) 
 ->  ( [. A  /  x ]. ps  ->  [. A  /  x ]. ch )
 )
 
Theoremsbctt 3054 Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/ x ph )  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbcgf 3055 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.)
 |- 
 F/ x ph   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc19.21g 3056 Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
 |- 
 F/ x ph   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps )  <->  ( ph  ->  [. A  /  x ]. ps ) ) )
 
Theoremsbcg 3057* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3055. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc2iegf 3058* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
 |- 
 F/ x ps   &    |-  F/ y ps   &    |-  F/ x  B  e.  W   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( [. A  /  x ].
 [. B  /  y ]. ph  <->  ps ) )
 
Theoremsbc2ie 3059* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
 
Theoremsbc2iedv 3060* Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ph  ->  ( ( x  =  A  /\  y  =  B )  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( [. A  /  x ]. [. B  /  y ]. ps  <->  ch ) )
 
Theoremsbc3ie 3061* Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  C  e.  _V   &    |-  (
 ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps ) )   =>    |-  ( [. A  /  x ]. [. B  /  y ]. [. C  /  z ]. ph  <->  ps )
 
Theoremsbccomlem 3062* Lemma for sbccom 3063. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom 3063* Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbcralt 3064* Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
 |-  ( ( A  e.  V  /\  F/_ y A ) 
 ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcrext 3065* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/_ y A ) 
 ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcralg 3066* Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcrexg 3067* Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. E. y  e.  B  ph  <->  E. y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcreug 3068* Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. E! y  e.  B  ph  <->  E! y  e.  B  [. A  /  x ]. ph ) )
 
Theoremsbcabel 3069* Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.)
 |-  F/_ x B   =>    |-  ( A  e.  V  ->  ( [. A  /  x ]. { y  | 
 ph }  e.  B  <->  { y  |  [. A  /  x ]. ph }  e.  B ) )
 
Theoremrspsbc 3070* 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 1969 and spsbc 3004. See also rspsbca 3071 and rspcsbela 3141. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( A  e.  B  ->  ( A. x  e.  B  ph  ->  [. A  /  x ]. ph )
 )
 
Theoremrspsbca 3071* Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.)
 |-  ( ( A  e.  B  /\  A. x  e.  B  ph )  ->  [. A  /  x ]. ph )
 
Theoremrspesbca 3072* Existence form of rspsbca 3071. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( ( A  e.  B  /\  [. A  /  x ].
 ph )  ->  E. x  e.  B  ph )
 
Theoremspesbc 3073 Existence form of spsbc 3004. (Contributed by Mario Carneiro, 18-Nov-2016.)
 |-  ( [. A  /  x ]. ph  ->  E. x ph )
 
Theoremspesbcd 3074 form of spsbc 3004. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  [. A  /  x ]. ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbcth2 3075* A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)
 |-  ( x  e.  B  -> 
 ph )   =>    |-  ( A  e.  B  -> 
 [. A  /  x ].
 ph )
 
Theoremra5 3076 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 1795. (Contributed by NM, 16-Jan-2004.)
 |- 
 F/ x ph   =>    |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( ph  ->  A. x  e.  A  ps ) )
 
Theoremrmo2 3077* Alternate definition of restricted "at most one." Note that  E* x  e.  A ph is not equivalent to  E. y  e.  A A. x  e.  A ( ph  ->  x  =  y ) (in analogy to reu6 2955); to see this, let  A be the empty set. However, one direction of this pattern holds; see rmo2i 3078. (Contributed by NM, 17-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E* x  e.  A ph  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
 
Theoremrmo2i 3078* Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E. y  e.  A  A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A ph )
 
Theoremrmo3 3079* Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E* x  e.  A ph  <->  A. x  e.  A  A. y  e.  A  ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
Theoremrmob 3080* Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  C  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( E* x  e.  A ph  /\  ( B  e.  A  /\  ps ) )  ->  ( B  =  C  <->  ( C  e.  A  /\  ch ) ) )
 
Theoremrmoi 3081* Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)
 |-  ( x  =  B  ->  ( ph  <->  ps ) )   &    |-  ( x  =  C  ->  (
 ph 
 <->  ch ) )   =>    |-  ( ( E* x  e.  A ph  /\  ( B  e.  A  /\  ps )  /\  ( C  e.  A  /\  ch ) )  ->  B  =  C )
 
2.1.10  Proper substitution of classes for sets into classes
 
Syntaxcsb 3082 Extend class notation to include the proper substitution of a class for a set into another class.
 class  [_ A  /  x ]_ B
 
Definitiondf-csb 3083* 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 2992, to prevent ambiguity. Theorem sbcel1g 3101 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3110 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
 
Theoremcsb2 3084* Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that  x can be free in  B but cannot occur in  A. (Contributed by NM, 2-Dec-2013.)
 |-  [_ A  /  x ]_ B  =  { y  |  E. x ( x  =  A  /\  y  e.  B ) }
 
Theoremcsbeq1 3085 Analog of dfsbcq 2994 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  =  B  -> 
 [_ A  /  x ]_ C  =  [_ B  /  x ]_ C )
 
Theoremcbvcsb 3086 Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on  A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)
 |-  F/_ y C   &    |-  F/_ x D   &    |-  ( x  =  y  ->  C  =  D )   =>    |-  [_ A  /  x ]_ C  =  [_ A  /  y ]_ D
 
Theoremcbvcsbv 3087* 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.)
 |-  ( x  =  y 
 ->  B  =  C )   =>    |-  [_ A  /  x ]_ B  =  [_ A  /  y ]_ C
 
Theoremcsbeq1d 3088 Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  [_ A  /  x ]_ C  =  [_ B  /  x ]_ C )
 
Theoremcsbid 3089 Analog of sbid 1865 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ x  /  x ]_ A  =  A
 
Theoremcsbeq1a 3090 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
 
Theoremcsbco 3091* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  [_ A  /  x ]_ B
 
Theoremcsbexg 3092 The existence of proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( ( A  e.  V  /\  A. x  B  e.  W )  ->  [_ A  /  x ]_ B  e.  _V )
 
Theoremcsbex 3093 The existence of proper substitution into a class. (Contributed by NM, 7-Aug-2007.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  [_ A  /  x ]_ B  e.  _V
 
Theoremcsbtt 3094 Substitution doesn't affect a constant  B (in which  x is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  ( ( A  e.  V  /\  F/_ x B ) 
 ->  [_ A  /  x ]_ B  =  B )
 
Theoremcsbconstgf 3095 Substitution doesn't affect a constant  B (in which  x is not free). (Contributed by NM, 10-Nov-2005.)
 |-  F/_ x B   =>    |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  B )
 
Theoremcsbconstg 3096* Substitution doesn't affect a constant  B (in which  x is not free). csbconstgf 3095 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  B )
 
Theoremsbcel12g 3097 Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e.  C  <->  [_ A  /  x ]_ B  e.  [_ A  /  x ]_ C ) )
 
Theoremsbceqg 3098 Distribute proper substitution through an equality relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =  C  <->  [_ A  /  x ]_ B  =  [_ A  /  x ]_ C ) )
 
Theoremsbcnel12g 3099 Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  e/  C  <->  [_ A  /  x ]_ B  e/  [_ A  /  x ]_ C ) )
 
Theoremsbcne12g 3100 Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  =/=  C  <->  [_ A  /  x ]_ B  =/=  [_ A  /  x ]_ C ) )
    < Previous  Next >

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 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31433
  Copyright terms: Public domain < Previous  Next >