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Type | Label | Description |
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Statement | ||
Theorem | isof1o 5801 | An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.) |
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Theorem | isorel 5802 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
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Theorem | isoresbr 5803* | A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) |
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Theorem | isoid 5804 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
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Theorem | isocnv 5805 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
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Theorem | isocnv2 5806 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
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Theorem | isores2 5807 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
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Theorem | isores1 5808 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
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Theorem | isores3 5809 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
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Theorem | isotr 5810 | Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
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Theorem | iso0 5811 |
The empty set is an ![]() ![]() ![]() |
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Theorem | isoini 5812 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
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Theorem | isoini2 5813 | Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
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Theorem | isoselem 5814* | Lemma for isose 5815. (Contributed by Mario Carneiro, 23-Jun-2015.) |
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Theorem | isose 5815 | An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
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Theorem | isopolem 5816 | Lemma for isopo 5817. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | isopo 5817 | An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | isosolem 5818 | Lemma for isoso 5819. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | isoso 5819 | An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
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Theorem | f1oiso 5820* |
Any one-to-one onto function determines an isomorphism with an induced
relation ![]() |
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Theorem | f1oiso2 5821* |
Any one-to-one onto function determines an isomorphism with an induced
relation ![]() |
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Theorem | canth 5822 |
No set ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Syntax | crio 5823 | Extend class notation with restricted description binder. |
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Definition | df-riota 5824 |
Define restricted description binder. In case there is no unique ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | riotaeqdv 5825* | Formula-building deduction for iota. (Contributed by NM, 15-Sep-2011.) |
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Theorem | riotabidv 5826* | Formula-building deduction for restricted iota. (Contributed by NM, 15-Sep-2011.) |
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Theorem | riotaeqbidv 5827* | Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.) |
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Theorem | riotaexg 5828* | Restricted iota is a set. (Contributed by Jim Kingdon, 15-Jun-2020.) |
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Theorem | riotav 5829 | An iota restricted to the universe is unrestricted. (Contributed by NM, 18-Sep-2011.) |
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Theorem | riotauni 5830 | Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.) |
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Theorem | nfriota1 5831* | The abstraction variable in a restricted iota descriptor isn't free. (Contributed by NM, 12-Oct-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | nfriotadxy 5832* | Deduction version of nfriota 5833. (Contributed by Jim Kingdon, 12-Jan-2019.) |
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Theorem | nfriota 5833* | A variable not free in a wff remains so in a restricted iota descriptor. (Contributed by NM, 12-Oct-2011.) |
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Theorem | cbvriota 5834* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | cbvriotav 5835* | Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | csbriotag 5836* | Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) |
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Theorem | riotacl2 5837 |
Membership law for "the unique element in ![]() ![]() (Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.) |
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Theorem | riotacl 5838* | Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
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Theorem | riotasbc 5839 | Substitution law for descriptions. (Contributed by NM, 23-Aug-2011.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
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Theorem | riotabidva 5840* | Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2725 analog.) (Contributed by NM, 17-Jan-2012.) |
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Theorem | riotabiia 5841 | Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2722 analog.) (Contributed by NM, 16-Jan-2012.) |
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Theorem | riota1 5842* | Property of restricted iota. Compare iota1 5187. (Contributed by Mario Carneiro, 15-Oct-2016.) |
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Theorem | riota1a 5843 | Property of iota. (Contributed by NM, 23-Aug-2011.) |
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Theorem | riota2df 5844* | A deduction version of riota2f 5845. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | riota2f 5845* |
This theorem shows a condition that allows us to represent a descriptor
with a class expression ![]() |
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Theorem | riota2 5846* |
This theorem shows a condition that allows us to represent a descriptor
with a class expression ![]() |
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Theorem | riotaprop 5847* | Properties of a restricted definite description operator. Todo (df-riota 5824 update): can some uses of riota2f 5845 be shortened with this? (Contributed by NM, 23-Nov-2013.) |
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Theorem | riota5f 5848* | A method for computing restricted iota. (Contributed by NM, 16-Apr-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
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Theorem | riota5 5849* | A method for computing restricted iota. (Contributed by NM, 20-Oct-2011.) (Revised by Mario Carneiro, 6-Dec-2016.) |
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Theorem | riotass2 5850* | Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.) |
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Theorem | riotass 5851* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.) |
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Theorem | moriotass 5852* | Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.) |
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Theorem | snriota 5853 | A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.) |
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Theorem | eusvobj2 5854* |
Specify the same property in two ways when class ![]() ![]() ![]() ![]() |
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Theorem | eusvobj1 5855* |
Specify the same object in two ways when class ![]() ![]() ![]() ![]() |
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Theorem | f1ofveu 5856* | There is one domain element for each value of a one-to-one onto function. (Contributed by NM, 26-May-2006.) |
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Theorem | f1ocnvfv3 5857* | Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
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Theorem | riotaund 5858* | Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.) |
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Theorem | acexmidlema 5859* | Lemma for acexmid 5867. (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmidlemb 5860* | Lemma for acexmid 5867. (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmidlemph 5861* | Lemma for acexmid 5867. (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmidlemab 5862* | Lemma for acexmid 5867. (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmidlemcase 5863* |
Lemma for acexmid 5867. Here we divide the proof into cases (based
on the
disjunction implicit in an unordered pair, not the sort of case
elimination which relies on excluded middle).
The cases are (1) the choice function evaluated at
Because of the way we represent the choice function
Although it isn't exactly about the division into cases, it is also
convenient for this lemma to also include the step that if the choice
function evaluated at (Contributed by Jim Kingdon, 7-Aug-2019.) |
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Theorem | acexmidlem1 5864* | Lemma for acexmid 5867. List the cases identified in acexmidlemcase 5863 and hook them up to the lemmas which handle each case. (Contributed by Jim Kingdon, 7-Aug-2019.) |
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Theorem | acexmidlem2 5865* |
Lemma for acexmid 5867. This builds on acexmidlem1 5864 by noting that every
element of ![]()
(Note that
The set (Contributed by Jim Kingdon, 5-Aug-2019.) |
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Theorem | acexmidlemv 5866* |
Lemma for acexmid 5867.
This is acexmid 5867 with additional disjoint variable conditions,
most
notably between (Contributed by Jim Kingdon, 6-Aug-2019.) |
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Theorem | acexmid 5867* |
The axiom of choice implies excluded middle. Theorem 1.3 in [Bauer]
p. 483.
The statement of the axiom of choice given here is ac2 in the Metamath
Proof Explorer (version of 3-Aug-2019). In particular, note that the
choice function Essentially the same proof can also be found at "The axiom of choice implies instances of EM", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". Often referred to as Diaconescu's theorem, or Diaconescu-Goodman-Myhill theorem, after Radu Diaconescu who discovered it in 1975 in the framework of topos theory and N. D. Goodman and John Myhill in 1978 in the framework of set theory (although it already appeared as an exercise in Errett Bishop's book Foundations of Constructive Analysis from 1967). For this theorem stated using the df-ac 7198 and df-exmid 4192 syntaxes, see exmidac 7201. (Contributed by Jim Kingdon, 4-Aug-2019.) |
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Syntax | co 5868 |
Extend class notation to include the value of an operation ![]() ![]() ![]() |
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Syntax | coprab 5869 | Extend class notation to include class abstraction (class builder) of nested ordered pairs. |
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Syntax | cmpo 5870 | Extend the definition of a class to include maps-to notation for defining an operation via a rule. |
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Definition | df-ov 5871 |
Define the value of an operation. Definition of operation value in
[Enderton] p. 79. Note that the syntax
is simply three class expressions
in a row bracketed by parentheses. There are no restrictions of any kind
on what those class expressions may be, although only certain kinds of
class expressions - a binary operation ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Definition | df-oprab 5872* |
Define the class abstraction (class builder) of a collection of nested
ordered pairs (for use in defining operations). This is a special case
of Definition 4.16 of [TakeutiZaring] p. 14. Normally ![]() ![]() ![]() |
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Definition | df-mpo 5873* |
Define maps-to notation for defining an operation via a rule. Read as
"the operation defined by the map from ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | oveq 5874 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq1 5875 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq2 5876 | Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq12 5877 | Equality theorem for operation value. (Contributed by NM, 16-Jul-1995.) |
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Theorem | oveq1i 5878 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq2i 5879 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) |
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Theorem | oveq12i 5880 | Equality inference for operation value. (Contributed by NM, 28-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
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Theorem | oveqi 5881 | Equality inference for operation value. (Contributed by NM, 24-Nov-2007.) |
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Theorem | oveq123i 5882 | Equality inference for operation value. (Contributed by FL, 11-Jul-2010.) |
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Theorem | oveq1d 5883 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
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Theorem | oveq2d 5884 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) |
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Theorem | oveqd 5885 | Equality deduction for operation value. (Contributed by NM, 9-Sep-2006.) |
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Theorem | oveq12d 5886 | Equality deduction for operation value. (Contributed by NM, 13-Mar-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
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Theorem | oveqan12d 5887 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
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Theorem | oveqan12rd 5888 | Equality deduction for operation value. (Contributed by NM, 10-Aug-1995.) |
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Theorem | oveq123d 5889 | Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.) |
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Theorem | fvoveq1d 5890 | Equality deduction for nested function and operation value. (Contributed by AV, 23-Jul-2022.) |
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Theorem | fvoveq1 5891 | Equality theorem for nested function and operation value. Closed form of fvoveq1d 5890. (Contributed by AV, 23-Jul-2022.) |
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Theorem | ovanraleqv 5892* | Equality theorem for a conjunction with an operation values within a restricted universal quantification. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 13-Aug-2022.) |
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Theorem | imbrov2fvoveq 5893 | Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.) |
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Theorem | ovrspc2v 5894* | If an operation value is element of a class for all operands of two classes, then the operation value is an element of the class for specific operands of the two classes. (Contributed by Mario Carneiro, 6-Dec-2014.) |
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Theorem | oveqrspc2v 5895* | Restricted specialization of operands, using implicit substitution. (Contributed by Mario Carneiro, 6-Dec-2014.) |
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Theorem | oveqdr 5896 | Equality of two operations for any two operands. Useful in proofs using *propd theorems. (Contributed by Mario Carneiro, 29-Jun-2015.) |
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Theorem | nfovd 5897 | Deduction version of bound-variable hypothesis builder nfov 5898. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
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Theorem | nfov 5898 | Bound-variable hypothesis builder for operation value. (Contributed by NM, 4-May-2004.) |
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Theorem | oprabidlem 5899* | Slight elaboration of exdistrfor 1800. A lemma for oprabid 5900. (Contributed by Jim Kingdon, 15-Jan-2019.) |
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Theorem | oprabid 5900 |
The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61.
Although this theorem would be useful with a distinct variable condition
between ![]() ![]() ![]() |
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