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Theorem List for Metamath Proof Explorer - 41401-41500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremofoacom 41401 Component-wise addition of natural numnber-yielding functions commutes. (Contributed by RP, 5-Jan-2025.)
((𝐴 ∈ 𝑉 ∧ (𝐹 ∈ (Ο‰ ↑m 𝐴) ∧ 𝐺 ∈ (Ο‰ ↑m 𝐴))) β†’ (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
 
Theoremnaddcnff 41402 Addition operator for Cantor normal forms is a function into Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) β†’ ( ∘f +o β†Ύ (𝑆 Γ— 𝑆)):(𝑆 Γ— 𝑆)βŸΆπ‘†)
 
Theoremnaddcnffn 41403 Addition operator for Cantor normal forms is a function. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) β†’ ( ∘f +o β†Ύ (𝑆 Γ— 𝑆)) Fn (𝑆 Γ— 𝑆))
 
Theoremnaddcnffo 41404 Addition of Cantor normal forms is a function onto Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) β†’ ( ∘f +o β†Ύ (𝑆 Γ— 𝑆)):(𝑆 Γ— 𝑆)–onto→𝑆)
 
Theoremnaddcnfcl 41405 Closure law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 2-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) β†’ (𝐹 ∘f +o 𝐺) ∈ 𝑆)
 
Theoremnaddcnfcom 41406 Component-wise ordinal addition of Cantor normal forms commutes. (Contributed by RP, 2-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆)) β†’ (𝐹 ∘f +o 𝐺) = (𝐺 ∘f +o 𝐹))
 
Theoremnaddcnfid1 41407 Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) β†’ (𝐹 ∘f +o (𝑋 Γ— {βˆ…})) = 𝐹)
 
Theoremnaddcnfid2 41408 Identity law for component-wise ordinal addition of Cantor normal forms. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ 𝐹 ∈ 𝑆) β†’ ((𝑋 Γ— {βˆ…}) ∘f +o 𝐹) = 𝐹)
 
Theoremnaddcnfass 41409 Component-wise addition of Cantor normal forms is associative. (Contributed by RP, 3-Jan-2025.)
(((𝑋 ∈ On ∧ 𝑆 = dom (Ο‰ CNF 𝑋)) ∧ (𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆)) β†’ ((𝐹 ∘f +o 𝐺) ∘f +o 𝐻) = (𝐹 ∘f +o (𝐺 ∘f +o 𝐻)))
 
21.32.2  Surreal Contributions
 
Theoremabeqabi 41410 Generalized condition for a class abstraction to be equal to some class. (Contributed by RP, 2-Sep-2024.)
𝐴 = {π‘₯ ∣ πœ“}    β‡’   ({π‘₯ ∣ πœ‘} = 𝐴 ↔ βˆ€π‘₯(πœ‘ ↔ πœ“))
 
Theoremabpr 41411* Condition for a class abstraction to be a pair. (Contributed by RP, 25-Aug-2024.)
({π‘₯ ∣ πœ‘} = {π‘Œ, 𝑍} ↔ βˆ€π‘₯(πœ‘ ↔ (π‘₯ = π‘Œ ∨ π‘₯ = 𝑍)))
 
Theoremabtp 41412* Condition for a class abstraction to be a triple. (Contributed by RP, 25-Aug-2024.)
({π‘₯ ∣ πœ‘} = {𝑋, π‘Œ, 𝑍} ↔ βˆ€π‘₯(πœ‘ ↔ (π‘₯ = 𝑋 ∨ π‘₯ = π‘Œ ∨ π‘₯ = 𝑍)))
 
Theoremralopabb 41413* Restricted universal quantification over an ordered-pair class abstraction. (Contributed by RP, 25-Sep-2024.)
𝑂 = {⟨π‘₯, π‘¦βŸ© ∣ πœ‘}    &   (π‘œ = ⟨π‘₯, π‘¦βŸ© β†’ (πœ“ ↔ πœ’))    β‡’   (βˆ€π‘œ ∈ 𝑂 πœ“ ↔ βˆ€π‘₯βˆ€π‘¦(πœ‘ β†’ πœ’))
 
Theorembropabg 41414* Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27047. (Contributed by RP, 26-Sep-2024.)
(π‘₯ = 𝐴 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝐡 β†’ (πœ“ ↔ πœ’))    &   π‘… = {⟨π‘₯, π‘¦βŸ© ∣ πœ‘}    β‡’   (𝐴𝑅𝐡 ↔ ((𝐴 ∈ V ∧ 𝐡 ∈ V) ∧ πœ’))
 
Theoremfpwfvss 41415 Functions into a powerset always have values which are subsets. This is dependant on our convention when the argument is not part of the domain. (Contributed by RP, 13-Sep-2024.)
𝐹:πΆβŸΆπ’« 𝐡    β‡’   (πΉβ€˜π΄) βŠ† 𝐡
 
Theoremsdomne0 41416 A class that strictly dominates any set is not empty. (Suggested by SN, 14-Jan-2025.) (Contributed by RP, 14-Jan-2025.)
(𝐡 β‰Ί 𝐴 β†’ 𝐴 β‰  βˆ…)
 
Theoremsdomne0d 41417 A class that strictly dominates any set is not empty. (Contributed by RP, 3-Sep-2024.)
(πœ‘ β†’ 𝐡 β‰Ί 𝐴)    &   (πœ‘ β†’ 𝐡 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝐴 β‰  βˆ…)
 
Theoremsafesnsupfiss 41418 If 𝐡 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
(πœ‘ β†’ (𝑂 = βˆ… ∨ 𝑂 = 1o))    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝑅 Or 𝐴)    β‡’   (πœ‘ β†’ if(𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡) βŠ† 𝐡)
 
Theoremsafesnsupfiub 41419* If 𝐡 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
(πœ‘ β†’ (𝑂 = βˆ… ∨ 𝑂 = 1o))    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝑅 Or 𝐴)    &   (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐢 π‘₯𝑅𝑦)    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ if (𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡)βˆ€π‘¦ ∈ 𝐢 π‘₯𝑅𝑦)
 
Theoremsafesnsupfidom1o 41420 If 𝐡 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 1-Sep-2024.)
(πœ‘ β†’ (𝑂 = βˆ… ∨ 𝑂 = 1o))    &   (πœ‘ β†’ 𝐡 ∈ Fin)    β‡’   (πœ‘ β†’ if(𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡) β‰Ό 1o)
 
Theoremsafesnsupfilb 41421* If 𝐡 is a finite subset of ordered class 𝐴, we can safely create a small subset with the same largest element and upper bound, if any. (Contributed by RP, 3-Sep-2024.)
(πœ‘ β†’ (𝑂 = βˆ… ∨ 𝑂 = 1o))    &   (πœ‘ β†’ 𝐡 ∈ Fin)    &   (πœ‘ β†’ 𝐡 βŠ† 𝐴)    &   (πœ‘ β†’ 𝑅 Or 𝐴)    β‡’   (πœ‘ β†’ βˆ€π‘₯ ∈ (𝐡 βˆ– if(𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡))βˆ€π‘¦ ∈ if (𝑂 β‰Ί 𝐡, {sup(𝐡, 𝐴, 𝑅)}, 𝐡)π‘₯𝑅𝑦)
 
Theoremisoeq145d 41422 Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.)
(πœ‘ β†’ 𝐹 = 𝐺)    &   (πœ‘ β†’ 𝐴 = 𝐢)    &   (πœ‘ β†’ 𝐡 = 𝐷)    β‡’   (πœ‘ β†’ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐡) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐢, 𝐷)))
 
Theoremresisoeq45d 41423 Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.)
(πœ‘ β†’ 𝐴 = 𝐢)    &   (πœ‘ β†’ 𝐡 = 𝐷)    β‡’   (πœ‘ β†’ ((𝐹 β†Ύ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐡) ↔ (𝐹 β†Ύ 𝐢) Isom 𝑅, 𝑆 (𝐢, 𝐷)))
 
Theoremnegslem1 41424 An equivalence between identically restricted order-reversing self-isometries. (Contributed by RP, 30-Sep-2024.)
(𝐴 = 𝐡 β†’ ((𝐹 β†Ύ 𝐴) Isom 𝑅, ◑𝑅(𝐴, 𝐴) ↔ (𝐹 β†Ύ 𝐡) Isom 𝑅, ◑𝑅(𝐡, 𝐡)))
 
Theoremnvocnvb 41425* Equivalence to saying the converse of an involution is the function itself. (Contributed by RP, 13-Oct-2024.)
((𝐹 Fn 𝐴 ∧ ◑𝐹 = 𝐹) ↔ (𝐹:𝐴–1-1-onto→𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (πΉβ€˜(πΉβ€˜π‘₯)) = π‘₯))
 
Theoremrp-brsslt 41426* Binary relation form of a relation, <, which has been extended from relation 𝑅 to subsets of class 𝑆. Usually, we will assume 𝑅 Or 𝑆. Definition in [Alling], p. 2. Generalization of brsslt 27047. (Originally by Scott Fenton, 8-Dec-2021.) (Contributed by RP, 28-Nov-2023.)
< = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž βŠ† 𝑆 ∧ 𝑏 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 π‘₯𝑅𝑦)}    β‡’   (𝐴 < 𝐡 ↔ ((𝐴 ∈ V ∧ 𝐡 ∈ V) ∧ (𝐴 βŠ† 𝑆 ∧ 𝐡 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 π‘₯𝑅𝑦)))
 
Theoremnla0002 41427* Extending a linear order to subsets, the empty set is less than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
< = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž βŠ† 𝑆 ∧ 𝑏 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 π‘₯𝑅𝑦)}    &   (πœ‘ β†’ 𝐴 ∈ V)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    β‡’   (πœ‘ β†’ βˆ… < 𝐴)
 
Theoremnla0003 41428* Extending a linear order to subsets, the empty set is greater than any subset. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
< = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž βŠ† 𝑆 ∧ 𝑏 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 π‘₯𝑅𝑦)}    &   (πœ‘ β†’ 𝐴 ∈ V)    &   (πœ‘ β†’ 𝐴 βŠ† 𝑆)    β‡’   (πœ‘ β†’ 𝐴 < βˆ…)
 
Theoremnla0001 41429* Extending a linear order to subsets, the empty set is less than itself. Note in [Alling], p. 3. (Contributed by RP, 28-Nov-2023.)
< = {βŸ¨π‘Ž, π‘βŸ© ∣ (π‘Ž βŠ† 𝑆 ∧ 𝑏 βŠ† 𝑆 ∧ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 π‘₯𝑅𝑦)}    β‡’   (πœ‘ β†’ βˆ… < βˆ…)
 
Theoremfaosnf0.11b 41430* 𝐡 is called a non-limit ordinal if it is not a limit ordinal. (Contributed by RP, 27-Sep-2023.)

Alling, Norman L. "Fundamentals of Analysis Over Surreal Numbers Fields." The Rocky Mountain Journal of Mathematics 19, no. 3 (1989): 565-73. http://www.jstor.org/stable/44237243.

((Ord 𝐴 ∧ Β¬ Lim 𝐴 ∧ 𝐴 β‰  βˆ…) β†’ βˆƒπ‘₯ ∈ On 𝐴 = suc π‘₯)
 
Theoremdfno2 41431 A surreal number, in the functional sign expansion representation, is a function which maps from an ordinal into a set of two possible signs. (Contributed by RP, 12-Jan-2025.)
No = {𝑓 ∈ 𝒫 (On Γ— {1o, 2o}) ∣ (Fun 𝑓 ∧ dom 𝑓 ∈ On)}
 
Theoremonnog 41432 Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
((𝐴 ∈ On ∧ 𝐡 ∈ {1o, 2o}) β†’ (𝐴 Γ— {𝐡}) ∈ No )
 
Theoremonnobdayg 41433 Every ordinal maps to a surreal number of that birthday. (Contributed by RP, 21-Sep-2023.)
((𝐴 ∈ On ∧ 𝐡 ∈ {1o, 2o}) β†’ ( bday β€˜(𝐴 Γ— {𝐡})) = 𝐴)
 
Theorembdaybndex 41434 Bounds formed from the birthday are surreal numbers. (Contributed by RP, 21-Sep-2023.)
((𝐴 ∈ No ∧ 𝐡 = ( bday β€˜π΄) ∧ 𝐢 ∈ {1o, 2o}) β†’ (𝐡 Γ— {𝐢}) ∈ No )
 
Theorembdaybndbday 41435 Bounds formed from the birthday have the same birthday. (Contributed by RP, 30-Sep-2023.)
((𝐴 ∈ No ∧ 𝐡 = ( bday β€˜π΄) ∧ 𝐢 ∈ {1o, 2o}) β†’ ( bday β€˜(𝐡 Γ— {𝐢})) = ( bday β€˜π΄))
 
Theoremonno 41436 Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(𝐴 ∈ On β†’ (𝐴 Γ— {2o}) ∈ No )
 
Theoremonnoi 41437 Every ordinal maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
𝐴 ∈ On    β‡’   (𝐴 Γ— {2o}) ∈ No
 
Theorem0no 41438 Ordinal zero maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
βˆ… ∈ No
 
Theorem1no 41439 Ordinal one maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(1o Γ— {2o}) ∈ No
 
Theorem2no 41440 Ordinal two maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(2o Γ— {2o}) ∈ No
 
Theorem3no 41441 Ordinal three maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(3o Γ— {2o}) ∈ No
 
Theorem4no 41442 Ordinal four maps to a surreal number. (Contributed by RP, 21-Sep-2023.)
(4o Γ— {2o}) ∈ No
 
Theoremfnimafnex 41443 The functional image of a function value exists. (Contributed by RP, 31-Oct-2024.)
𝐹 Fn 𝐡    β‡’   (𝐹 β€œ (πΊβ€˜π΄)) ∈ V
 
21.32.3  Short Studies
 
Theoremnlimsuc 41444 A successor is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
(𝐴 ∈ On β†’ Β¬ Lim suc 𝐴)
 
Theoremnlim1NEW 41445 1 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
Β¬ Lim 1o
 
Theoremnlim2NEW 41446 2 is not a limit ordinal. (Contributed by BTernaryTau, 1-Dec-2024.) (Proof shortened by RP, 13-Dec-2024.)
Β¬ Lim 2o
 
Theoremnlim3 41447 3 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
Β¬ Lim 3o
 
Theoremnlim4 41448 4 is not a limit ordinal. (Contributed by RP, 13-Dec-2024.)
Β¬ Lim 4o
 
Theoremoa1un 41449 Given 𝐴 ∈ On, let 𝐴 +o 1o be defined to be the union of 𝐴 and {𝐴}. Compare with oa1suc 8445. (Contributed by RP, 27-Sep-2023.)
(𝐴 ∈ On β†’ (𝐴 +o 1o) = (𝐴 βˆͺ {𝐴}))
 
Theoremoa1cl 41450 𝐴 +o 1o is in On. (Contributed by RP, 27-Sep-2023.)
(𝐴 ∈ On β†’ (𝐴 +o 1o) ∈ On)
 
Theorem0finon 41451 0 is a finite ordinal. See peano1 7816. (Contributed by RP, 27-Sep-2023.)
βˆ… ∈ (On ∩ Fin)
 
Theorem1finon 41452 1 is a finite ordinal. See 1onn 8554. (Contributed by RP, 27-Sep-2023.)
1o ∈ (On ∩ Fin)
 
Theorem2finon 41453 2 is a finite ordinal. See 1onn 8554. (Contributed by RP, 27-Sep-2023.)
2o ∈ (On ∩ Fin)
 
Theorem3finon 41454 3 is a finite ordinal. See 1onn 8554. (Contributed by RP, 27-Sep-2023.)
3o ∈ (On ∩ Fin)
 
Theorem4finon 41455 4 is a finite ordinal. See 1onn 8554. (Contributed by RP, 27-Sep-2023.)
4o ∈ (On ∩ Fin)
 
Theoremfinona1cl 41456 The finite ordinals are closed under the add one operation. (Contributed by RP, 27-Sep-2023.)
(𝑁 ∈ (On ∩ Fin) β†’ (𝑁 +o 1o) ∈ (On ∩ Fin))
 
Theoremfinonex 41457 The finite ordinals are a set. See also onprc 7703 and fiprc 8923 for proof that On and Fin are proper classes. (Contributed by RP, 27-Sep-2023.)
(On ∩ Fin) ∈ V
 
Theoremfzunt 41458 Union of two adjacent finite sets of sequential integers that share a common endpoint. (Suggested by NM, 21-Jul-2005.) (Contributed by RP, 14-Dec-2024.)
(((𝐾 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝐾 ≀ 𝑀 ∧ 𝑀 ≀ 𝑁)) β†’ ((𝐾...𝑀) βˆͺ (𝑀...𝑁)) = (𝐾...𝑁))
 
Theoremfzuntd 41459 Union of two adjacent finite sets of sequential integers that share a common endpoint. (Contributed by RP, 14-Dec-2024.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ≀ 𝑀)    &   (πœ‘ β†’ 𝑀 ≀ 𝑁)    β‡’   (πœ‘ β†’ ((𝐾...𝑀) βˆͺ (𝑀...𝑁)) = (𝐾...𝑁))
 
Theoremfzunt1d 41460 Union of two overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝐿 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ≀ 𝑀)    &   (πœ‘ β†’ 𝑀 ≀ 𝐿)    &   (πœ‘ β†’ 𝐿 ≀ 𝑁)    β‡’   (πœ‘ β†’ ((𝐾...𝐿) βˆͺ (𝑀...𝑁)) = (𝐾...𝑁))
 
Theoremfzuntgd 41461 Union of two adjacent or overlapping finite sets of sequential integers. (Contributed by RP, 14-Dec-2024.)
(πœ‘ β†’ 𝐾 ∈ β„€)    &   (πœ‘ β†’ 𝐿 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝐾 ≀ 𝑀)    &   (πœ‘ β†’ 𝑀 ≀ (𝐿 + 1))    &   (πœ‘ β†’ 𝐿 ≀ 𝑁)    β‡’   (πœ‘ β†’ ((𝐾...𝐿) βˆͺ (𝑀...𝑁)) = (𝐾...𝑁))
 
21.32.3.1  Additional work on conditional logical operator
 
Theoremifpan123g 41462 Conjunction of conditional logical operators. (Contributed by RP, 18-Apr-2020.)
((if-(πœ‘, πœ’, 𝜏) ∧ if-(πœ“, πœƒ, πœ‚)) ↔ (((Β¬ πœ‘ ∨ πœ’) ∧ (πœ‘ ∨ 𝜏)) ∧ ((Β¬ πœ“ ∨ πœƒ) ∧ (πœ“ ∨ πœ‚))))
 
Theoremifpan23 41463 Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)
((if-(πœ‘, πœ“, πœ’) ∧ if-(πœ‘, πœƒ, 𝜏)) ↔ if-(πœ‘, (πœ“ ∧ πœƒ), (πœ’ ∧ 𝜏)))
 
Theoremifpdfor2 41464 Define or in terms of conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ∨ πœ“) ↔ if-(πœ‘, πœ‘, πœ“))
 
Theoremifporcor 41465 Corollary of commutation of or. (Contributed by RP, 20-Apr-2020.)
(if-(πœ‘, πœ‘, πœ“) ↔ if-(πœ“, πœ“, πœ‘))
 
Theoremifpdfan2 41466 Define and with conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((πœ‘ ∧ πœ“) ↔ if-(πœ‘, πœ“, πœ‘))
 
Theoremifpancor 41467 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(πœ‘, πœ“, πœ‘) ↔ if-(πœ“, πœ‘, πœ“))
 
Theoremifpdfor 41468 Define or in terms of conditional logic operator and true. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ∨ πœ“) ↔ if-(πœ‘, ⊀, πœ“))
 
Theoremifpdfan 41469 Define and with conditional logic operator and false. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ∧ πœ“) ↔ if-(πœ‘, πœ“, βŠ₯))
 
Theoremifpbi2 41470 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((πœ‘ ↔ πœ“) β†’ (if-(πœ’, πœ‘, πœƒ) ↔ if-(πœ’, πœ“, πœƒ)))
 
Theoremifpbi3 41471 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((πœ‘ ↔ πœ“) β†’ (if-(πœ’, πœƒ, πœ‘) ↔ if-(πœ’, πœƒ, πœ“)))
 
Theoremifpim1 41472 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ‘ β†’ πœ“) ↔ if-(Β¬ πœ‘, ⊀, πœ“))
 
Theoremifpnot 41473 Restate negated wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
(Β¬ πœ‘ ↔ if-(πœ‘, βŠ₯, ⊀))
 
Theoremifpid2 41474 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
(πœ‘ ↔ if-(πœ‘, ⊀, βŠ₯))
 
Theoremifpim2 41475 Restate implication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ‘ β†’ πœ“) ↔ if-(πœ“, ⊀, Β¬ πœ‘))
 
Theoremifpbi23 41476 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((πœ‘ ↔ πœ“) ∧ (πœ’ ↔ πœƒ)) β†’ (if-(𝜏, πœ‘, πœ’) ↔ if-(𝜏, πœ“, πœƒ)))
 
Theoremifpbiidcor 41477 Restatement of biid 261. (Contributed by RP, 25-Apr-2020.)
if-(πœ‘, πœ‘, Β¬ πœ‘)
 
Theoremifpbicor 41478 Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.)
(if-(πœ‘, πœ“, Β¬ πœ“) ↔ if-(πœ“, πœ‘, Β¬ πœ‘))
 
Theoremifpxorcor 41479 Corollary of commutation of biconditional. (Contributed by RP, 25-Apr-2020.)
(if-(πœ‘, Β¬ πœ“, πœ“) ↔ if-(πœ“, Β¬ πœ‘, πœ‘))
 
Theoremifpbi1 41480 Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.)
((πœ‘ ↔ πœ“) β†’ (if-(πœ‘, πœ’, πœƒ) ↔ if-(πœ“, πœ’, πœƒ)))
 
Theoremifpnot23 41481 Negation of conditional logical operator. (Contributed by RP, 18-Apr-2020.)
(Β¬ if-(πœ‘, πœ“, πœ’) ↔ if-(πœ‘, Β¬ πœ“, Β¬ πœ’))
 
Theoremifpnotnotb 41482 Factor conditional logic operator over negation in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)
(if-(πœ‘, Β¬ πœ“, Β¬ πœ’) ↔ Β¬ if-(πœ‘, πœ“, πœ’))
 
Theoremifpnorcor 41483 Corollary of commutation of nor. (Contributed by RP, 25-Apr-2020.)
(if-(πœ‘, Β¬ πœ‘, Β¬ πœ“) ↔ if-(πœ“, Β¬ πœ“, Β¬ πœ‘))
 
Theoremifpnancor 41484 Corollary of commutation of and. (Contributed by RP, 25-Apr-2020.)
(if-(πœ‘, Β¬ πœ“, Β¬ πœ‘) ↔ if-(πœ“, Β¬ πœ‘, Β¬ πœ“))
 
Theoremifpnot23b 41485 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(Β¬ if-(πœ‘, Β¬ πœ“, πœ’) ↔ if-(πœ‘, πœ“, Β¬ πœ’))
 
Theoremifpbiidcor2 41486 Restatement of biid 261. (Contributed by RP, 25-Apr-2020.)
Β¬ if-(πœ‘, Β¬ πœ‘, πœ‘)
 
Theoremifpnot23c 41487 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(Β¬ if-(πœ‘, πœ“, Β¬ πœ’) ↔ if-(πœ‘, Β¬ πœ“, πœ’))
 
Theoremifpnot23d 41488 Negation of conditional logical operator. (Contributed by RP, 25-Apr-2020.)
(Β¬ if-(πœ‘, Β¬ πœ“, Β¬ πœ’) ↔ if-(πœ‘, πœ“, πœ’))
 
Theoremifpdfnan 41489 Define nand as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ⊼ πœ“) ↔ if-(πœ‘, Β¬ πœ“, ⊀))
 
Theoremifpdfxor 41490 Define xor as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ⊻ πœ“) ↔ if-(πœ‘, Β¬ πœ“, πœ“))
 
Theoremifpbi12 41491 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((πœ‘ ↔ πœ“) ∧ (πœ’ ↔ πœƒ)) β†’ (if-(πœ‘, πœ’, 𝜏) ↔ if-(πœ“, πœƒ, 𝜏)))
 
Theoremifpbi13 41492 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((πœ‘ ↔ πœ“) ∧ (πœ’ ↔ πœƒ)) β†’ (if-(πœ‘, 𝜏, πœ’) ↔ if-(πœ“, 𝜏, πœƒ)))
 
Theoremifpbi123 41493 Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.)
(((πœ‘ ↔ πœ“) ∧ (πœ’ ↔ πœƒ) ∧ (𝜏 ↔ πœ‚)) β†’ (if-(πœ‘, πœ’, 𝜏) ↔ if-(πœ“, πœƒ, πœ‚)))
 
Theoremifpidg 41494 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœƒ ↔ if-(πœ‘, πœ“, πœ’)) ↔ ((((πœ‘ ∧ πœ“) β†’ πœƒ) ∧ ((πœ‘ ∧ πœƒ) β†’ πœ“)) ∧ ((πœ’ β†’ (πœ‘ ∨ πœƒ)) ∧ (πœƒ β†’ (πœ‘ ∨ πœ’)))))
 
Theoremifpid3g 41495 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ’ ↔ if-(πœ‘, πœ“, πœ’)) ↔ (((πœ‘ ∧ πœ“) β†’ πœ’) ∧ ((πœ‘ ∧ πœ’) β†’ πœ“)))
 
Theoremifpid2g 41496 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ“ ↔ if-(πœ‘, πœ“, πœ’)) ↔ ((πœ“ β†’ (πœ‘ ∨ πœ’)) ∧ (πœ’ β†’ (πœ‘ ∨ πœ“))))
 
Theoremifpid1g 41497 Restate wff as conditional logic operator. (Contributed by RP, 20-Apr-2020.)
((πœ‘ ↔ if-(πœ‘, πœ“, πœ’)) ↔ ((πœ’ β†’ πœ‘) ∧ (πœ‘ β†’ πœ“)))
 
Theoremifpim23g 41498 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(((πœ‘ β†’ πœ“) ↔ if-(πœ’, πœ“, Β¬ πœ‘)) ↔ (((πœ‘ ∧ πœ“) β†’ πœ’) ∧ (πœ’ β†’ (πœ‘ ∨ πœ“))))
 
Theoremifpim3 41499 Restate implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
((πœ‘ β†’ πœ“) ↔ if-(πœ‘, πœ“, Β¬ πœ‘))
 
Theoremifpnim1 41500 Restate negated implication as conditional logic operator. (Contributed by RP, 25-Apr-2020.)
(Β¬ (πœ‘ β†’ πœ“) ↔ if-(πœ‘, Β¬ πœ“, πœ‘))
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