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Theorem List for Metamath Proof Explorer - 3001-3100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsbc2or 3001* 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  [ A  /  x ] ph behavior at proper classes, matching the sbc5 3017 (false) and sbc6 3019 (true) conclusions. This is interesting since dfsbcq 2995 and dfsbcq2 2996 (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  y that  ph or  A may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.)
 |-  ( ( [. A  /  x ]. ph  <->  E. x ( x  =  A  /\  ph )
 )  \/  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
 ) )
 
Theoremsbcex 3002 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 3003 Equality theorem for class substitution. Class version of sbequ12 1862. (Contributed by NM, 26-Sep-2003.)
 |-  ( x  =  A  ->  ( ph  <->  [. A  /  x ].
 ph ) )
 
Theoremsbceq2a 3004 Equality theorem for class substitution. Class version of sbequ12r 1863. (Contributed by NM, 4-Jan-2017.)
 |-  ( A  =  x 
 ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremspsbc 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 1966 and rspsbc 3071. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( A. x ph  -> 
 [. A  /  x ].
 ph ) )
 
Theoremspsbcd 3006 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 1966 and rspsbc 3071. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A. x ps )   =>    |-  ( ph  ->  [. A  /  x ]. ps )
 
Theoremsbcth 3007 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 3008* Deduction version of sbcth 3007. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  A  e.  V )  ->  [. A  /  x ]. ps )
 
Theoremsbcid 3009 An identity theorem for substitution. See sbid 1865. (Contributed by Mario Carneiro, 18-Feb-2017.)
 |-  ( [. x  /  x ]. ph  <->  ph )
 
Theoremnfsbc1d 3010 Deduction version of nfsbc1 3011. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/ x [. A  /  x ]. ps )
 
Theoremnfsbc1 3011 Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |-  F/_ x A   =>    |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbc1v 3012* Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.)
 |- 
 F/ x [. A  /  x ]. ph
 
Theoremnfsbcd 3013 Deduction version of nfsbc 3014. (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 3014 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 3015* 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 3016* 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 3017* 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 3018* 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 3019* 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 3020* An equivalence for class substitution in the spirit of df-clab 2272. 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 3021 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 3022* 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 3023* Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3024.) (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 3024* 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 3025* 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 3026* Conversion of implicit substitution to explicit class substitution. This version of sbcie 3027 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 3027* 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 3028* 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 3029* 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 3030* 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 3031 Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2924 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 3032* Substitution applied to an atomic wff. Set theory version of eqsb3 2386. (Contributed by Andrew Salmon, 29-Jun-2011.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. x  =  B  <->  A  =  B ) )
 
Theoremsbcng 3033 Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)
 |-  ( A  e.  V  ->  ( [. A  /  x ].  -.  ph  <->  -.  [. A  /  x ].
 ph ) )
 
Theoremsbcimg 3034 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 3035 Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( [. A  /  x ]. ph  /\  [. A  /  x ]. ps )
 )
 
Theoremsbcang 3036 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 3037 Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.)
 |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( [. A  /  x ]. ph  \/  [. A  /  x ]. ps ) )
 
Theoremsbcorg 3038 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 3039 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 3040* 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 3041* 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 3042* 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 3043* 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 3044* Set theory version of sbeqal1 27608. (Contributed by Andrew Salmon, 28-Jun-2011.)
 |-  ( A  e.  V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B )
 )
 
Theoremsbeqalb 3045* 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 3046 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 3047* Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 [. A  /  x ].
 ps 
 <-> 
 [. A  /  x ].
 ch ) )
 
Theoremsbcbii 3048 Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)
 |-  ( ph  <->  ps )   =>    |-  ( [. A  /  x ]. ph  <->  [. A  /  x ].
 ps )
 
TheoremsbcbiiOLD 3049 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 3050* eqsbc3 3032 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 28689 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
 |-  ( A  e.  B  ->  ( [. A  /  x ]. C  =  x  <->  C  =  A )
 )
 
Theoremsbc3ang 3051 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 3052* 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 3053* 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 3054* 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 3055 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 3056 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 3057 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 3058* Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3056. (Contributed by Alan Sare, 10-Nov-2012.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
 
Theoremsbc2iegf 3059* 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 3060* 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 3061* 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 3062* 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 3063* Lemma for sbccom 3064. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.)
 |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom 3064* 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 3065* 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 3066* 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 3067* 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 3068* 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 3069* 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 3070* 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 3071* 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 1966 and spsbc 3005. See also rspsbca 3072 and rspcsbela 3142. (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 3072* 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 3073* Existence form of rspsbca 3072. (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 3074 Existence form of spsbc 3005. (Contributed by Mario Carneiro, 18-Nov-2016.)
 |-  ( [. A  /  x ]. ph  ->  E. x ph )
 
Theoremspesbcd 3075 form of spsbc 3005. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  [. A  /  x ]. ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbcth2 3076* 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 3077 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 3078* 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 2956); to see this, let  A be the empty set. However, one direction of this pattern holds; see rmo2i 3079. (Contributed by NM, 17-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E* x  e.  A ph  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
 
Theoremrmo2i 3079* 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 3080* 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 3081* 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 3082* 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 3083 Extend class notation to include the proper substitution of a class for a set into another class.
 class  [_ A  /  x ]_ B
 
Definitiondf-csb 3084* 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 2993, to prevent ambiguity. Theorem sbcel1g 3102 shows an example of how ambiguity could arise if we didn't use distinguished brackets. Theorem sbccsbg 3111 recreates substitution into a wff from this definition. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
 
Theoremcsb2 3085* 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 3086 Analog of dfsbcq 2995 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  =  B  -> 
 [_ A  /  x ]_ C  =  [_ B  /  x ]_ C )
 
Theoremcbvcsb 3087 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 3088* 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 3089 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 3090 Analog of sbid 1865 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ x  /  x ]_ A  =  A
 
Theoremcsbeq1a 3091 Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)
 |-  ( x  =  A  ->  B  =  [_ A  /  x ]_ B )
 
Theoremcsbco 3092* Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)
 |-  [_ A  /  y ]_ [_ y  /  x ]_ B  =  [_ A  /  x ]_ B
 
Theoremcsbexg 3093 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 3094 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 3095 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 3096 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 3097* Substitution doesn't affect a constant  B (in which  x is not free). csbconstgf 3096 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ B  =  B )
 
Theoremsbcel12g 3098 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 3099 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 3100 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 ) )
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