P. 37 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 3601-3700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ifbieq12i 3601	Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
|- ( ph <-> ps )   &    |- A = C   &    |- B = D   =>    |- if ( ph , A , B ) = if ( ps , C , D )
Theorem	ifbieq12d 3602	Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> A = C )   &    |- ( ph -> B = D )   =>    |- ( ph -> if ( ps , A , B ) = if ( ch , C , D ) )
Theorem	nfifd 3603	Deduction version of nfif 3604. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- ( ph -> F/ x ps )   &    |- ( ph -> F/_ x A )   &    |- ( ph -> F/_ x B )   =>    |- ( ph -> F/_ x if ( ps , A , B ) )
Theorem	nfif 3604	Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- F/ x ph   &    |- F/_ x A   &    |- F/_ x B   =>    |- F/_ x if ( ph , A , B )
Theorem	ifcldadc 3605	Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.)
|- ( ( ph /\ ps ) -> A e. C )   &    |- ( ( ph /\ -. ps ) -> B e. C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) e. C )
Theorem	ifeq1dadc 3606	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ ps ) -> A = B )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , C ) = if ( ps , B , C ) )
Theorem	ifeq2dadc 3607	Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( ph /\ -. ps ) -> A = B )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , C , A ) = if ( ps , C , B ) )
Theorem	ifeqdadc 3608	Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
|- ( ( ph /\ ps ) -> A = C )   &    |- ( ( ph /\ -. ps ) -> B = C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) = C )
Theorem	ifbothdadc 3609	A formula th containing a decidable conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 3-Jun-2022.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   &    |- ( ( et /\ ph ) -> ps )   &    |- ( ( et /\ -. ph ) -> ch )   &    |- ( et -> DECID ph )   =>    |- ( et -> th )
Theorem	ifbothdc 3610	A wff th containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( A = if ( ph , A , B ) -> ( ps <-> th ) )   &    |- ( B = if ( ph , A , B ) -> ( ch <-> th ) )   =>    |- ( ( ps /\ ch /\ DECID ph ) -> th )
Theorem	ifiddc 3611	Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
|- (DECID ph -> if ( ph , A , A ) = A )
Theorem	eqifdc 3612	Expansion of an equality with a conditional operator. (Contributed by Jim Kingdon, 28-Jul-2022.)
|- (DECID ph -> ( A = if ( ph , B , C ) <-> ( ( ph /\ A = B ) \/ ( -. ph /\ A = C ) ) ) )
Theorem	ifcldcd 3613	Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
|- ( ph -> A e. C )   &    |- ( ph -> B e. C )   &    |- ( ph -> DECID ps )   =>    |- ( ph -> if ( ps , A , B ) e. C )
Theorem	ifnotdc 3614	Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
|- (DECID ph -> if ( -. ph , A , B ) = if ( ph , B , A ) )
Theorem	ifandc 3615	Rewrite a conjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- (DECID ph -> if ( ( ph /\ ps ) , A , B ) = if ( ph , if ( ps , A , B ) , B ) )
Theorem	ifordc 3616	Rewrite a disjunction in a conditional as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- (DECID ph -> if ( ( ph \/ ps ) , A , B ) = if ( ph , A , if ( ps , A , B ) ) )
Theorem	ifmdc 3617	If a conditional class is inhabited, then the condition is decidable. This shows that conditionals are not very useful unless one can prove the condition decidable. (Contributed by BJ, 24-Sep-2022.)
|- ( A e. if ( ph , B , C ) -> DECID ph )
Theorem	ifnetruedc 3618	Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( (DECID ph /\ A =/= B /\ if ( ph , A , B ) = A ) -> ph )
Theorem	ifnefals 3619	Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( ( A =/= B /\ if ( ph , A , B ) = B ) -> -. ph )
Theorem	ifnebibdc 3620	The converse of ifbi 3596 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
|- ( (DECID ph /\ DECID ps /\ A =/= B ) -> ( if ( ph , A , B ) = if ( ps , A , B ) <-> ( ph <-> ps ) ) )
2.1.16  Power classes
Syntax	cpw 3621	Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class ~P A
Theorem	pwjust 3622*	Soundness justification theorem for df-pw 3623. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- { x | x C_ A } = { y | y C_ A }
Definition	df-pw 3623*	Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of _V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if A is { 3 , 5 , 7 }, then ~P A is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.)
|- ~P A = { x | x C_ A }
Theorem	pweq 3624	Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> ~P A = ~P B )
Theorem	pweqi 3625	Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
|- A = B   =>    |- ~P A = ~P B
Theorem	pweqd 3626	Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
|- ( ph -> A = B )   =>    |- ( ph -> ~P A = ~P B )
Theorem	elpw 3627	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
|- A e. _V   =>    |- ( A e. ~P B <-> A C_ B )
Theorem	velpw 3628*	Setvar variable membership in a power class (common case). See elpw 3627. (Contributed by David A. Wheeler, 8-Dec-2018.)
|- ( x e. ~P A <-> x C_ A )
Theorem	elpwg 3629	Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
|- ( A e. V -> ( A e. ~P B <-> A C_ B ) )
Theorem	elpwi 3630	Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
|- ( A e. ~P B -> A C_ B )
Theorem	elpwb 3631	Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)
|- ( A e. ~P B <-> ( A e. _V /\ A C_ B ) )
Theorem	elpwid 3632	An element of a power class is a subclass. Deduction form of elpwi 3630. (Contributed by David Moews, 1-May-2017.)
|- ( ph -> A e. ~P B )   =>    |- ( ph -> A C_ B )
Theorem	elelpwi 3633	If A belongs to a part of C then A belongs to C. (Contributed by FL, 3-Aug-2009.)
|- ( ( A e. B /\ B e. ~P C ) -> A e. C )
Theorem	nfpw 3634	Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
|- F/_ x A   =>    |- F/_ x ~P A
Theorem	pwidg 3635	Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
|- ( A e. V -> A e. ~P A )
Theorem	pwid 3636	A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
|- A e. _V   =>    |- A e. ~P A
Theorem	pwss 3637*	Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
|- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) )
2.1.17  Unordered and ordered pairs
Syntax	csn 3638	Extend class notation to include singleton.
class { A }
Syntax	cpr 3639	Extend class notation to include unordered pair.
class { A , B }
Syntax	ctp 3640	Extend class notation to include unordered triplet.
class { A , B , C }
Syntax	cop 3641	Extend class notation to include ordered pair.
class <. A , B >.
Syntax	cotp 3642	Extend class notation to include ordered triple.
class <. A , B , C >.
Theorem	snjust 3643*	Soundness justification theorem for df-sn 3644. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- { x | x = A } = { y | y = A }
Definition	df-sn 3644*	Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of _V, although it is not very meaningful in this case. For an alternate definition see dfsn2 3652. (Contributed by NM, 5-Aug-1993.)
|- { A } = { x | x = A }
Definition	df-pr 3645	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. They are unordered, so { A , B } = { B , A } as proven by prcom 3714. For a more traditional definition, but requiring a dummy variable, see dfpr2 3657. (Contributed by NM, 5-Aug-1993.)
|- { A , B } = ( { A } u. { B } )
Definition	df-tp 3646	Define unordered triple of classes. Definition of [Enderton] p. 19. (Contributed by NM, 9-Apr-1994.)
|- { A , B , C } = ( { A , B } u. { C } )
Definition	df-op 3647*	Definition of an ordered pair, equivalent to Kuratowski's definition { { A } , { A , B } } when the arguments are sets. Since the behavior of Kuratowski definition is not very useful for proper classes, we define it to be empty in this case (see opprc1 3847 and opprc2 3848). For Kuratowski's actual definition when the arguments are sets, see dfop 3824. Definition 9.1 of [Quine] p. 58 defines an ordered pair unconditionally as <. A , B >. = { { A } , { A , B } }, which has different behavior from our df-op 3647 when the arguments are proper classes. Ordinarily this difference is not important, since neither definition is meaningful in that case. Our df-op 3647 was chosen because it often makes proofs shorter by eliminating unnecessary sethood hypotheses. There are other ways to define ordered pairs. The basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition <. A , B >.2 = { { { A } , (/) } , { { B } } }. This was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition is <. A , B >.3 = { A , { A , B } }, but it requires the Axiom of Regularity for its justification and is not commonly used. Finally, an ordered pair of real numbers can be represented by a complex number. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
|- <. A , B >. = { x | ( A e. _V /\ B e. _V /\ x e. { { A } , { A , B } } ) }
Definition	df-ot 3648	Define ordered triple of classes. Definition of ordered triple in [Stoll] p. 25. (Contributed by NM, 3-Apr-2015.)
|- <. A , B , C >. = <. <. A , B >. , C >.
Theorem	sneq 3649	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.)
|- ( A = B -> { A } = { B } )
Theorem	sneqi 3650	Equality inference for singletons. (Contributed by NM, 22-Jan-2004.)
|- A = B   =>    |- { A } = { B }
Theorem	sneqd 3651	Equality deduction for singletons. (Contributed by NM, 22-Jan-2004.)
|- ( ph -> A = B )   =>    |- ( ph -> { A } = { B } )
Theorem	dfsn2 3652	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A } = { A , A }
Theorem	elsng 3653	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
|- ( A e. V -> ( A e. { B } <-> A = B ) )
Theorem	elsn 3654	There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
|- A e. _V   =>    |- ( A e. { B } <-> A = B )
Theorem	velsn 3655	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. (Contributed by NM, 21-Jun-1993.)
|- ( x e. { A } <-> x = A )
Theorem	elsni 3656	There is only one element in a singleton. (Contributed by NM, 5-Jun-1994.)
|- ( A e. { B } -> A = B )
Theorem	dfpr2 3657*	Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)
|- { A , B } = { x | ( x = A \/ x = B ) }
Theorem	elprg 3658	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
|- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
Theorem	elpr 3659	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
|- A e. _V   =>    |- ( A e. { B , C } <-> ( A = B \/ A = C ) )
Theorem	elpr2 3660	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.)
|- B e. _V   &    |- C e. _V   =>    |- ( A e. { B , C } <-> ( A = B \/ A = C ) )
Theorem	elpri 3661	If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
|- ( A e. { B , C } -> ( A = B \/ A = C ) )
Theorem	nelpri 3662	If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
|- A =/= B   &    |- A =/= C   =>    |- -. A e. { B , C }
Theorem	prneli 3663	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using e/. (Contributed by David A. Wheeler, 10-May-2015.)
|- A =/= B   &    |- A =/= C   =>    |- A e/ { B , C }
Theorem	nelprd 3664	If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
|- ( ph -> A =/= B )   &    |- ( ph -> A =/= C )   =>    |- ( ph -> -. A e. { B , C } )
Theorem	eldifpr 3665	Membership in a set with two elements removed. Similar to eldifsn 3766 and eldiftp 3684. (Contributed by Mario Carneiro, 18-Jul-2017.)
|- ( A e. ( B \ { C , D } ) <-> ( A e. B /\ A =/= C /\ A =/= D ) )
Theorem	rexdifpr 3666	Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.)
|- ( E. x e. ( A \ { B , C } ) ph <-> E. x e. A ( x =/= B /\ x =/= C /\ ph ) )
Theorem	snidg 3667	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)
|- ( A e. V -> A e. { A } )
Theorem	snidb 3668	A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)
|- ( A e. _V <-> A e. { A } )
Theorem	snid 3669	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)
|- A e. _V   =>    |- A e. { A }
Theorem	vsnid 3670	A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
|- x e. { x }
Theorem	elsn2g 3671	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 28-Oct-2003.)
|- ( B e. V -> ( A e. { B } <-> A = B ) )
Theorem	elsn2 3672	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 12-Jun-1994.)
|- B e. _V   =>    |- ( A e. { B } <-> A = B )
Theorem	nelsn 3673	If a class is not equal to the class in a singleton, then it is not in the singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by BJ, 4-May-2021.)
|- ( A =/= B -> -. A e. { B } )
Theorem	mosn 3674*	A singleton has at most one element. This works whether A is a proper class or not, and in that sense can be seen as encompassing both snmg 3756 and snprc 3703. (Contributed by Jim Kingdon, 30-Aug-2018.)
|- E* x x e. { A }
Theorem	ralsnsg 3675*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
Theorem	ralsns 3676*	Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )
Theorem	rexsns 3677*	Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
|- ( E. x e. { A } ph <-> [. A / x ]. ph )
Theorem	ralsng 3678*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( A. x e. { A } ph <-> ps ) )
Theorem	rexsng 3679*	Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
|- ( x = A -> ( ph <-> ps ) )   =>    |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )
Theorem	exsnrex 3680	There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)
|- ( E. x M = { x } <-> E. x e. M M = { x } )
Theorem	ralsn 3681*	Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( A. x e. { A } ph <-> ps )
Theorem	rexsn 3682*	Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
|- A e. _V   &    |- ( x = A -> ( ph <-> ps ) )   =>    |- ( E. x e. { A } ph <-> ps )
Theorem	eltpg 3683	Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
|- ( A e. V -> ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) )
Theorem	eldiftp 3684	Membership in a set with three elements removed. Similar to eldifsn 3766 and eldifpr 3665. (Contributed by David A. Wheeler, 22-Jul-2017.)
|- ( A e. ( B \ { C , D , E } ) <-> ( A e. B /\ ( A =/= C /\ A =/= D /\ A =/= E ) ) )
Theorem	eltpi 3685	A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
|- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) )
Theorem	eltp 3686	A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)
|- A e. _V   =>    |- ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) )
Theorem	dftp2 3687*	Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)
|- { A , B , C } = { x | ( x = A \/ x = B \/ x = C ) }
Theorem	nfpr 3688	Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   &    |- F/_ x B   =>    |- F/_ x { A , B }
Theorem	ralprg 3689*	Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )
Theorem	rexprg 3690*	Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
Theorem	raltpg 3691*	Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   &    |- ( x = C -> ( ph <-> th ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) )
Theorem	rextpg 3692*	Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   &    |- ( x = C -> ( ph <-> th ) )   =>    |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) )
Theorem	ralpr 3693*	Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( A. x e. { A , B } ph <-> ( ps /\ ch ) )
Theorem	rexpr 3694*	Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   =>    |- ( E. x e. { A , B } ph <-> ( ps \/ ch ) )
Theorem	raltp 3695*	Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   &    |- ( x = C -> ( ph <-> th ) )   =>    |- ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) )
Theorem	rextp 3696*	Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x = A -> ( ph <-> ps ) )   &    |- ( x = B -> ( ph <-> ch ) )   &    |- ( x = C -> ( ph <-> th ) )   =>    |- ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) )
Theorem	sbcsng 3697*	Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
|- ( A e. V -> ( [. A / x ]. ph <-> A. x e. { A } ph ) )
Theorem	nfsn 3698	Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)
|- F/_ x A   =>    |- F/_ x { A }
Theorem	csbsng 3699	Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
|- ( A e. V -> [_ A / x ]_ { B } = { [_ A / x ]_ B } )
Theorem	disjsn 3700	Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)
|- ( ( A i^i { B } ) = (/) <-> -. B e. A )