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Theorem List for Intuitionistic Logic Explorer - 901-1000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm3.11dc 901 Theorem *3.11 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.1 704, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))))
 
Theorempm3.12dc 902 Theorem *3.12 of [WhiteheadRussell] p. 111, but for decidable propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))))
 
Theorempm3.13dc 903 Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 703, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.)
(DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))))
 
Theoremdn1dc 904 DN1 for decidable propositions. Without the decidability conditions, DN1 can serve as a single axiom for Boolean algebra. See http://www-unix.mcs.anl.gov/~mccune/papers/basax/v12.pdf. (Contributed by Jim Kingdon, 22-Apr-2018.)
((DECID 𝜑 ∧ (DECID 𝜓 ∧ (DECID 𝜒DECID 𝜃))) → (¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒))
 
Theorempm5.71dc 905 Decidable proposition version of theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for decidability by Jim Kingdon, 19-Apr-2018.)
(DECID 𝜓 → ((𝜓 → ¬ 𝜒) → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒))))
 
Theorempm5.75 906 Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.)
(((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒))
 
Theorembimsc1 907 Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)
(((𝜑𝜓) ∧ (𝜒 ↔ (𝜓𝜑))) → (𝜒𝜑))
 
Theoremccase 908 Inference for combining cases. (Contributed by NM, 29-Jul-1999.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)
((𝜑𝜓) → 𝜏)    &   ((𝜒𝜓) → 𝜏)    &   ((𝜑𝜃) → 𝜏)    &   ((𝜒𝜃) → 𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 
Theoremccased 909 Deduction for combining cases. (Contributed by NM, 9-May-2004.)
(𝜑 → ((𝜓𝜒) → 𝜂))    &   (𝜑 → ((𝜃𝜒) → 𝜂))    &   (𝜑 → ((𝜓𝜏) → 𝜂))    &   (𝜑 → ((𝜃𝜏) → 𝜂))       (𝜑 → (((𝜓𝜃) ∧ (𝜒𝜏)) → 𝜂))
 
Theoremccase2 910 Inference for combining cases. (Contributed by NM, 29-Jul-1999.)
((𝜑𝜓) → 𝜏)    &   (𝜒𝜏)    &   (𝜃𝜏)       (((𝜑𝜒) ∧ (𝜓𝜃)) → 𝜏)
 
Theoremniabn 911 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → ((𝜒𝜓) ↔ ¬ 𝜑))
 
Theoremdedlem0a 912 Alternate version of dedlema 913. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
(𝜑 → (𝜓 ↔ ((𝜒𝜑) → (𝜓𝜑))))
 
Theoremdedlema 913 Lemma for iftrue 3384. (Contributed by NM, 26-Jun-2002.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑 → (𝜓 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 
Theoremdedlemb 914 Lemma for iffalse 3387. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 7-May-2011.)
𝜑 → (𝜒 ↔ ((𝜓𝜑) ∨ (𝜒 ∧ ¬ 𝜑))))
 
Theorempm4.42r 915 One direction of Theorem *4.42 of [WhiteheadRussell] p. 119. (Contributed by Jim Kingdon, 4-Aug-2018.)
(((𝜑𝜓) ∨ (𝜑 ∧ ¬ 𝜓)) → 𝜑)
 
Theoremninba 916 Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.)
𝜑       𝜓 → (¬ 𝜑 ↔ (𝜒𝜓)))
 
Theoremprlem1 917 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 918 A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
 
Theoremoplem1 919 A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))    &   (𝜓𝜃)    &   (𝜒 → (𝜃𝜏))       (𝜑𝜓)
 
Theoremrnlem 920 Lemma used in construction of real numbers. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (((𝜑𝜒) ∧ (𝜓𝜃)) ∧ ((𝜑𝜃) ∧ (𝜓𝜒))))
 
1.2.12  Abbreviated conjunction and disjunction of three wff's
 
Syntaxw3o 921 Extend wff definition to include 3-way disjunction ('or').
wff (𝜑𝜓𝜒)
 
Syntaxw3a 922 Extend wff definition to include 3-way conjunction ('and').
wff (𝜑𝜓𝜒)
 
Definitiondf-3or 923 Define disjunction ('or') of 3 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 717. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
 
Definitiondf-3an 924 Define conjunction ('and') of 3 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 393. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
 
Theorem3orass 925 Associative law for triple disjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))
 
Theorem3anass 926 Associative law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑 ∧ (𝜓𝜒)))
 
Theorem3anrot 927 Rotation law for triple conjunction. (Contributed by NM, 8-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))
 
Theorem3orrot 928 Rotation law for triple disjunction. (Contributed by NM, 4-Apr-1995.)
((𝜑𝜓𝜒) ↔ (𝜓𝜒𝜑))
 
Theorem3ancoma 929 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜓𝜑𝜒))
 
Theorem3ancomb 930 Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))
 
Theorem3orcomb 931 Commutation law for triple disjunction. (Contributed by Scott Fenton, 20-Apr-2011.)
((𝜑𝜓𝜒) ↔ (𝜑𝜒𝜓))
 
Theorem3anrev 932 Reversal law for triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) ↔ (𝜒𝜓𝜑))
 
Theorem3anan32 933 Convert triple conjunction to conjunction, then commute. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜒) ∧ 𝜓))
 
Theorem3anan12 934 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 935 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜑𝜒)))
 
Theoremanandi3r 936 Distribution of triple conjunction over conjunction. (Contributed by David A. Wheeler, 4-Nov-2018.)
((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ (𝜒𝜓)))
 
Theorem3ioran 937 Negated triple disjunction as triple conjunction. (Contributed by Scott Fenton, 19-Apr-2011.)
(¬ (𝜑𝜓𝜒) ↔ (¬ 𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜒))
 
Theorem3simpa 938 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → (𝜑𝜓))
 
Theorem3simpb 939 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → (𝜑𝜒))
 
Theorem3simpc 940 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((𝜑𝜓𝜒) → (𝜓𝜒))
 
Theoremsimp1 941 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜑)
 
Theoremsimp2 942 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜓)
 
Theoremsimp3 943 Simplification of triple conjunction. (Contributed by NM, 21-Apr-1994.)
((𝜑𝜓𝜒) → 𝜒)
 
Theoremsimpl1 944 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜑)
 
Theoremsimpl2 945 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimpl3 946 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
(((𝜑𝜓𝜒) ∧ 𝜃) → 𝜒)
 
Theoremsimpr1 947 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜓)
 
Theoremsimpr2 948 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜒)
 
Theoremsimpr3 949 Simplification rule. (Contributed by Jeff Hankins, 17-Nov-2009.)
((𝜑 ∧ (𝜓𝜒𝜃)) → 𝜃)
 
Theoremsimp1i 950 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜑
 
Theoremsimp2i 951 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜓
 
Theoremsimp3i 952 Infer a conjunct from a triple conjunction. (Contributed by NM, 19-Apr-2005.)
(𝜑𝜓𝜒)       𝜒
 
Theoremsimp1d 953 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theoremsimp2d 954 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜒)
 
Theoremsimp3d 955 Deduce a conjunct from a triple conjunction. (Contributed by NM, 4-Sep-2005.)
(𝜑 → (𝜓𝜒𝜃))       (𝜑𝜃)
 
Theoremsimp1bi 956 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜓)
 
Theoremsimp2bi 957 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜒)
 
Theoremsimp3bi 958 Deduce a conjunct from a triple conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑 ↔ (𝜓𝜒𝜃))       (𝜑𝜃)
 
Theorem3adant1 959 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜃𝜑𝜓) → 𝜒)
 
Theorem3adant2 960 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜃𝜓) → 𝜒)
 
Theorem3adant3 961 Deduction adding a conjunct to antecedent. (Contributed by NM, 16-Jul-1995.)
((𝜑𝜓) → 𝜒)       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3ad2ant1 962 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜑𝜓𝜃) → 𝜒)
 
Theorem3ad2ant2 963 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜑𝜃) → 𝜒)
 
Theorem3ad2ant3 964 Deduction adding conjuncts to an antecedent. (Contributed by NM, 21-Apr-2005.)
(𝜑𝜒)       ((𝜓𝜃𝜑) → 𝜒)
 
Theoremsimp1l 965 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜑)
 
Theoremsimp1r 966 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
(((𝜑𝜓) ∧ 𝜒𝜃) → 𝜓)
 
Theoremsimp2l 967 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜓)
 
Theoremsimp2r 968 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑 ∧ (𝜓𝜒) ∧ 𝜃) → 𝜒)
 
Theoremsimp3l 969 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜒)
 
Theoremsimp3r 970 Simplification of triple conjunction. (Contributed by NM, 9-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃)) → 𝜃)
 
Theoremsimp11 971 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜑)
 
Theoremsimp12 972 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜓)
 
Theoremsimp13 973 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
(((𝜑𝜓𝜒) ∧ 𝜃𝜏) → 𝜒)
 
Theoremsimp21 974 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimp22 975 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜒)
 
Theoremsimp23 976 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑 ∧ (𝜓𝜒𝜃) ∧ 𝜏) → 𝜃)
 
Theoremsimp31 977 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜒)
 
Theoremsimp32 978 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜃)
 
Theoremsimp33 979 Simplification of doubly triple conjunction. (Contributed by NM, 17-Nov-2011.)
((𝜑𝜓 ∧ (𝜒𝜃𝜏)) → 𝜏)
 
Theoremsimpll1 980 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜑)
 
Theoremsimpll2 981 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimpll3 982 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜒)
 
Theoremsimplr1 983 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜑)
 
Theoremsimplr2 984 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜓)
 
Theoremsimplr3 985 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜃 ∧ (𝜑𝜓𝜒)) ∧ 𝜏) → 𝜒)
 
Theoremsimprl1 986 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜑)
 
Theoremsimprl2 987 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜓)
 
Theoremsimprl3 988 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓𝜒) ∧ 𝜃)) → 𝜒)
 
Theoremsimprr1 989 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜑)
 
Theoremsimprr2 990 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜓)
 
Theoremsimprr3 991 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜃 ∧ (𝜑𝜓𝜒))) → 𝜒)
 
Theoremsimpl1l 992 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏) → 𝜑)
 
Theoremsimpl1r 993 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((((𝜑𝜓) ∧ 𝜒𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimpl2l 994 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜑)
 
Theoremsimpl2r 995 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒 ∧ (𝜑𝜓) ∧ 𝜃) ∧ 𝜏) → 𝜓)
 
Theoremsimpl3l 996 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜑)
 
Theoremsimpl3r 997 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
(((𝜒𝜃 ∧ (𝜑𝜓)) ∧ 𝜏) → 𝜓)
 
Theoremsimpr1l 998 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜑)
 
Theoremsimpr1r 999 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ ((𝜑𝜓) ∧ 𝜒𝜃)) → 𝜓)
 
Theoremsimpr2l 1000 Simplification of conjunction. (Contributed by NM, 9-Mar-2012.)
((𝜏 ∧ (𝜒 ∧ (𝜑𝜓) ∧ 𝜃)) → 𝜑)
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