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Theorem List for Metamath Proof Explorer - 1001-1100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdedlem0b 1001 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.)
𝜑 → (𝜓 ↔ ((𝜓𝜑) → (𝜒𝜑))))

Theoremdedlema 1002 Lemma for weak deduction theorem. See also ifptru 1023. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Theoremdedlemb 1003 Lemma for weak deduction theorem. See also ifpfal 1024. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))

Theorempm4.42 1004 Theorem *4.42 of [WhiteheadRussell] p. 119. See also ifpid 1025. (Contributed by Roy F. Longton, 21-Jun-2005.)
(𝜑 ↔ ((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))

Theoremprlem1 1005 A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
(𝜑 → (𝜂𝜒))    &   (𝜓 → ¬ 𝜃)       (𝜑 → (𝜓 → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂)))

Theoremprlem2 1006 A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))

Theoremoplem1 1007 A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜓𝜃)    &   (𝜒 → (𝜃𝜏))       (𝜑𝜓)

Theoremdn1 1008 A single axiom for Boolean algebra known as DN1. See McCune, Veroff, Fitelson, Harris, Feist, Wos, Short single axioms for Boolean algebra, Journal of Automated Reasoning, 29(1):1--16, 2002. (https://www.cs.unm.edu/~mccune/papers/basax/v12.pdf). (Contributed by Jeff Hankins, 3-Jul-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
(¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒)

Theorembianir 1009 A closed form of mpbir 221, analogous to pm2.27 42 (assertion). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Roger Witte, 17-Aug-2020.)
((𝜑 ∧ (𝜓𝜑)) → 𝜓)

Theoremjaoi2 1010 Inference removing a negated conjunct in a disjunction of an antecedent if this conjunct is part of the disjunction. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)
((𝜑 ∨ (¬ 𝜑𝜒)) → 𝜓)       ((𝜑𝜒) → 𝜓)

Theoremjaoi3 1011 Inference separating a disjunct of an antecedent. (Contributed by Alexander van der Vekens, 25-May-2018.)
(𝜑𝜓)    &   ((¬ 𝜑𝜒) → 𝜓)       ((𝜑𝜒) → 𝜓)

1.2.8  The conditional operator for propositions

This subsection introduces the conditional operator for propositions, denoted by if- (see df-ifp 1013). It is the analogue for propositions of the conditional operator for classes if (see df-if 4085).

Syntaxwif 1012 Extend class notation to include the conditional operator for propositions.
wff if-(𝜑, 𝜓, 𝜒)

Definitiondf-ifp 1013 Definition of the conditional operator for propositions. The value of if-(𝜑, 𝜓, 𝜒) is 𝜓 if 𝜑 is true and 𝜒 if 𝜑 false. See dfifp2 1014, dfifp3 1015, dfifp4 1016, dfifp5 1017, dfifp6 1018 and dfifp7 1019 for alternate definitions.

This definition (in the form of dfifp2 1014) appears in Section II.24 of [Church] p. 129 (Definition D12 page 132), where it is called "conditioned disjunction". Church's [𝜓, 𝜑, 𝜒] corresponds to our if-(𝜑, 𝜓, 𝜒) (note the permutation of the first two variables).

Church uses the conditional operator as an intermediate step to prove completeness of some systems of connectives. The first result is that the system {if-, ⊤, ⊥} is complete: for the induction step, consider a wff with n+1 variables; single out one variable, say 𝜑; when one sets 𝜑 to True (resp. False), then what remains is a wff of n variables, so by the induction hypothesis it corresponds to a formula using only {if-, ⊤, ⊥}, say 𝜓 (resp. 𝜒); therefore, the formula if-(𝜑, 𝜓, 𝜒) represents the initial wff. Now, since { → , ¬ } and similar systems suffice to express if-, ⊤, ⊥, they are also complete.

(Contributed by BJ, 22-Jun-2019.)

(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))

Theoremdfifp2 1014 Alternate definition of the conditional operator for propositions. The value of if-(𝜑, 𝜓, 𝜒) is "if 𝜑 then 𝜓, and if not 𝜑 then 𝜒." This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 1013). (Contributed by BJ, 22-Jun-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Theoremdfifp3 1015 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremdfifp4 1016 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (𝜑𝜒)))

Theoremdfifp5 1017 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((¬ 𝜑𝜓) ∧ (¬ 𝜑𝜒)))

Theoremdfifp6 1018 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ ¬ (𝜒𝜑)))

Theoremdfifp7 1019 Alternate definition of the conditional operator for propositions. (Contributed by BJ, 2-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ ((𝜒𝜑) → (𝜑𝜓)))

Theoremanifp 1020 The conditional operator is implied by the conjunction of its possible outputs. Dual statement of ifpor 1021. (Contributed by BJ, 30-Sep-2019.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))

Theoremifpor 1021 The conditional operator implies the disjunction of its possible outputs. Dual statement of anifp 1020. (Contributed by BJ, 1-Oct-2019.)
(if-(𝜑, 𝜓, 𝜒) → (𝜓𝜒))

Theoremifpn 1022 Conditional operator for the negation of a proposition. (Contributed by BJ, 30-Sep-2019.)
(if-(𝜑, 𝜓, 𝜒) ↔ if-(¬ 𝜑, 𝜒, 𝜓))

Theoremifptru 1023 Value of the conditional operator for propositions when its first argument is true. Analogue for propositions of iftrue 4090. This is essentially dedlema 1002. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 10-Jul-2020.)
(𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜓))

Theoremifpfal 1024 Value of the conditional operator for propositions when its first argument is false. Analogue for propositions of iffalse 4093. This is essentially dedlemb 1003. (Contributed by BJ, 20-Sep-2019.) (Proof shortened by Wolf Lammen, 25-Jun-2020.)
𝜑 → (if-(𝜑, 𝜓, 𝜒) ↔ 𝜒))

Theoremifpid 1025 Value of the conditional operator for propositions when the same proposition is returned in either case. Analogue for propositions of ifid 4123. This is essentially pm4.42 1004. (Contributed by BJ, 20-Sep-2019.)
(if-(𝜑, 𝜓, 𝜓) ↔ 𝜓)

Theoremcasesifp 1026 Version of cases 992 expressed using if-. Case disjunction according to the value of 𝜑. One can see this as a proof that the two hypotheses characterize the conditional operator for propositions. For the converses, see ifptru 1023 and ifpfal 1024. (Contributed by BJ, 20-Sep-2019.)
(𝜑 → (𝜓𝜒))    &   𝜑 → (𝜓𝜃))       (𝜓 ↔ if-(𝜑, 𝜒, 𝜃))

Theoremifpbi123d 1027 Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.)
(𝜑 → (𝜓𝜏))    &   (𝜑 → (𝜒𝜂))    &   (𝜑 → (𝜃𝜁))       (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))

Theoremifpimpda 1028 Separation of the values of the conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 27-Feb-2021.)
((𝜑𝜓) → 𝜒)    &   ((𝜑 ∧ ¬ 𝜓) → 𝜃)       (𝜑 → if-(𝜓, 𝜒, 𝜃))

Theorem1fpid3 1029 The value of the conditional operator for propositions is its third argument if the first and second argument imply the third argument. (Contributed by AV, 4-Apr-2021.)
((𝜑𝜓) → 𝜒)       (if-(𝜑, 𝜓, 𝜒) → 𝜒)

1.2.9  The weak deduction theorem

This subsection contains a few results related to the weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html.

Theoremelimh 1030 Hypothesis builder for the weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.)
((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜒𝜏))    &   ((if-(𝜒, 𝜑, 𝜓) ↔ 𝜓) → (𝜃𝜏))    &   𝜃       𝜏

Theoremdedt 1031 The weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.)
((if-(𝜒, 𝜑, 𝜓) ↔ 𝜑) → (𝜃𝜏))    &   𝜏       (𝜒𝜃)

Theoremcon3ALT 1032 Proof of con3 149 from its associated inference con3i 150 that illustrates the use of the weak deduction theorem dedt 1031. (Contributed by NM, 27-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

TheoremelimhOLD 1033 Old version of elimh 1030. Obsolete as of 16-Mar-2021. (Contributed by NM, 26-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ ((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜒𝜏))    &   ((𝜓 ↔ ((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜃𝜏))    &   𝜃       𝜏

TheoremdedtOLD 1034 Old version of dedt 1031. Obsolete as of 16-Mar-2021. (Contributed by NM, 26-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ↔ ((𝜑𝜒) ∨ (𝜓 ∧ ¬ 𝜒))) → (𝜃𝜏))    &   𝜏       (𝜒𝜃)

Theoremcon3OLD 1035 Old version of con3ALT 1032. Obsolete as of 16-Mar-2021. (Contributed by NM, 27-Jun-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))

1.2.10  Abbreviated conjunction and disjunction of three wff's

Syntaxw3o 1036 Extend wff definition to include three-way disjunction ('or').
wff (𝜑𝜓𝜒)

Syntaxw3a 1037 Extend wff definition to include three-way conjunction ('and').
wff (𝜑𝜓𝜒)

Definitiondf-3or 1038 Define disjunction ('or') of three wff's. Definition *2.33 of [WhiteheadRussell] p. 105. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law orass 546. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))

Definitiondf-3an 1039 Define conjunction ('and') of three wff's. Definition *4.34 of [WhiteheadRussell] p. 118. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law anass 681. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))

Theorem3orass 1040 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))

Theorem3anass 1041 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))

Theorem3anrot 1042 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))

Theorem3orrot 1043 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))

Theorem3ancoma 1044 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))

Theorem3orcoma 1045 Commutation law for triple disjunction. (Contributed by Mario Carneiro, 4-Sep-2016.)
((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))

Theorem3ancomb 1046 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))

Theorem3orcomb 1047 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))

Theorem3anrev 1048 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜒𝜓𝜑))

Theorem3anan32 1049 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))

Theorem3anan12 1050 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
((𝜑𝜓𝜒) ↔ (𝜓 ∧ (𝜑𝜒)))

Theoremanandi3 1051 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))

Theoremanandi3r 1052 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))

Theorem3anor 1053 Triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.)
((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Theorem3ianor 1054 Negated triple conjunction expressed in terms of triple disjunction. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∨ ¬ 𝜓 ∨ ¬ 𝜒))

Theorem3ioran 1055 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))

Theorem3oran 1056 Triple disjunction in terms of triple conjunction. (Contributed by NM, 8-Oct-2012.)
((𝜑𝜓𝜒) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))

Theorem3simpa 1057 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → (𝜑𝜓))

Theorem3simpb 1058 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → (𝜑𝜒))

Theorem3simpc 1059 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((𝜑𝜓𝜒) → (𝜓𝜒))

Theoremsimp1 1060 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜑)

Theoremsimp2 1061 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜓)

Theoremsimp3 1062 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜒)

Theoremsimpl1 1063 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜑)

Theoremsimpl2 1064 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜓)

Theoremsimpl3 1065 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜒)

Theoremsimpr1 1066 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜓)

Theoremsimpr2 1067 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜒)

Theoremsimpr3 1068 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜃)

Theoremsimp1i 1069 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜑

Theoremsimp2i 1070 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜓

Theoremsimp3i 1071 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜒

Theoremsimp1d 1072 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)

Theoremsimp2d 1073 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜒)

Theoremsimp3d 1074 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜃)

Theoremsimp1bi 1075 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜓)

Theoremsimp2bi 1076 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜒)

Theoremsimp3bi 1077 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜃)

Theorem3adant1 1078 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜃𝜑𝜓) → 𝜒)

Theorem3adant2 1079 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜃𝜓) → 𝜒)

Theorem3adant3 1080 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜓𝜃) → 𝜒)

Theorem3ad2ant1 1081 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜑𝜓𝜃) → 𝜒)

Theorem3ad2ant2 1082 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜑𝜃) → 𝜒)

Theorem3ad2ant3 1083 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜃𝜑) → 𝜒)

Theoremsimp1l 1084 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜑)

Theoremsimp1r 1085 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜓)

Theoremsimp2l 1086 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜓)

Theoremsimp2r 1087 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜒)

Theoremsimp3l 1088 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜒)

Theoremsimp3r 1089 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜃)

Theoremsimp11 1090 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜑)

Theoremsimp12 1091 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜓)

Theoremsimp13 1092 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜒)

Theoremsimp21 1093 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜓)

Theoremsimp22 1094 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜒)

Theoremsimp23 1095 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜃)

Theoremsimp31 1096 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜒)

Theoremsimp32 1097 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜃)

Theoremsimp33 1098 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜏)

Theoremsimpll1 1099 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)

Theoremsimpll2 1100 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)

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