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Theorem List for Intuitionistic Logic Explorer - 801-900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-stab 801 Propositions where a double-negative can be removed are called stable. See Chapter 2 [Moschovakis] p. 2.

Our notation for stability is a connective STAB which we place before the formula in question. For example, STAB  x  =  y corresponds to " x  =  y is stable".

(Contributed by David A. Wheeler, 13-Aug-2018.)

 |-  (STAB 
 ph 
 <->  ( -.  -.  ph  -> 
 ph ) )
 
Theoremstbid 802 The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (STAB  ps  <-> STAB  ch )
 )
 
Theoremstabnot 803 Every negated formula is stable. (Contributed by David A. Wheeler, 13-Aug-2018.)
 |- STAB  -.  ph
 
1.2.8  Decidable propositions
 
Syntaxwdc 804 Extend wff definition to include decidability.
 wff DECID  ph
 
Definitiondf-dc 805 Propositions which are known to be true or false are called decidable. The (classical) Law of the Excluded Middle corresponds to the principle that all propositions are decidable, but even given intuitionistic logic, particular kinds of propositions may be decidable (for example, the proposition that two natural numbers are equal will be decidable under most sets of axioms).

Our notation for decidability is a connective DECID which we place before the formula in question. For example, DECID  x  =  y corresponds to " x  =  y is decidable".

We could transform intuitionistic logic to classical logic by adding unconditional forms of condc 823, exmiddc 806, peircedc 884, or notnotrdc 813, any of which would correspond to the assertion that all propositions are decidable.

(Contributed by Jim Kingdon, 11-Mar-2018.)

 |-  (DECID 
 ph 
 <->  ( ph  \/  -.  ph ) )
 
Theoremexmiddc 806 Law of excluded middle, for a decidable proposition. The law of the excluded middle is also called the principle of tertium non datur. Theorem *2.11 of [WhiteheadRussell] p. 101. It says that something is either true or not true; there are no in-between values of truth. The key way in which intuitionistic logic differs from classical logic is that intuitionistic logic says that excluded middle only holds for some propositions, and classical logic says that it holds for all propositions. (Contributed by Jim Kingdon, 12-May-2018.)
 |-  (DECID 
 ph  ->  ( ph  \/  -.  ph ) )
 
Theorempm2.1dc 807 Commuted law of the excluded middle for a decidable proposition. Based on theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by Jim Kingdon, 25-Mar-2018.)
 |-  (DECID 
 ph  ->  ( -.  ph  \/  ph ) )
 
Theoremdcbid 808 Equivalence property for decidability. Deduction form. (Contributed by Jim Kingdon, 7-Sep-2019.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (DECID  ps  <-> DECID  ch )
 )
 
Theoremdcbiit 809 Equivalence property for decidability. Closed form. (Contributed by BJ, 27-Jan-2020.)
 |-  ( ( ph  <->  ps )  ->  (DECID  ph  <-> DECID  ps ) )
 
Theoremdcbii 810 Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.)
 |-  ( ph  <->  ps )   =>    |-  (DECID 
 ph 
 <-> DECID  ps )
 
Theoremdcim 811 An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  -> DECID 
 ( ph  ->  ps )
 ) )
 
Theoremdcn 812 The negation of a decidable proposition is decidable. The converse need not hold, but does hold for negated propositions, see dcnn 818. (Contributed by Jim Kingdon, 25-Mar-2018.)
 |-  (DECID 
 ph  -> DECID  -.  ph )
 
Theoremnotnotrdc 813 Double negation elimination for a decidable proposition. The converse, notnot 603, holds for all propositions, not just decidable ones. This is Theorem *2.14 of [WhiteheadRussell] p. 102, but with a decidability condition added. (Contributed by Jim Kingdon, 11-Mar-2018.)
 |-  (DECID 
 ph  ->  ( -.  -.  ph 
 ->  ph ) )
 
Theoremdcstab 814 Decidability implies stability. The converse need not hold. (Contributed by David A. Wheeler, 13-Aug-2018.)
 |-  (DECID 
 ph  -> STAB  ph )
 
Theoremstdcndc 815 A formula is decidable if and only if its negation is decidable and it is stable (that is, it is testable and stable). (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof shortened by BJ, 28-Oct-2023.)
 |-  ( (STAB 
 ph  /\ DECID  -.  ph )  <-> DECID  ph )
 
TheoremstdcndcOLD 816 Obsolete version of stdcndc 815 as of 28-Oct-2023. (Contributed by David A. Wheeler, 13-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( (STAB 
 ph  /\ DECID  -.  ph )  <-> DECID  ph )
 
Theoremstdcn 817 A formula is stable if and only if the decidability of its negation implies its decidability. Note that the right-hand side of this biconditional is the converse of dcn 812. (Contributed by BJ, 18-Nov-2023.)
 |-  (STAB 
 ph 
 <->  (DECID 
 -.  ph  -> DECID  ph ) )
 
Theoremdcnn 818 Decidability of the negation of a proposition is equivalent to decidability of its double negation. See also dcn 812. The relation between dcn 812 and dcnn 818 is analogous to that between notnot 603 and notnotnot 608 (and directly stems from it). Using the notion of "testable proposition" (proposition whose negation is decidable), dcnn 818 means that a proposition is testable if and only if its negation is testable, and dcn 812 means that decidability implies testability. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof shortened by BJ, 25-Nov-2023.)
 |-  (DECID 
 -.  ph  <-> DECID  -.  -.  ph )
 
TheoremdcnnOLD 819 Obsolete proof of dcnnOLD 819 as of 25-Nov-2023. (Contributed by David A. Wheeler, 6-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (DECID 
 -.  ph  <-> DECID  -.  -.  ph )
 
Theoremnnexmid 820 Double negation of excluded middle. Intuitionistic logic refutes the negation of excluded middle (but does not prove excluded middle) for any formula. Can also be proved quickly from bj-nnor 12842 as in bj-nndcALT 12859. (Contributed by BJ, 9-Oct-2019.)
 |- 
 -.  -.  ( ph  \/  -.  ph )
 
Theoremnndc 821 Double negation of decidability of a formula. Intuitionistic logic refutes undecidability (but does not prove decidability) of any formula. (Contributed by BJ, 9-Oct-2019.)
 |- 
 -.  -. DECID  ph
 
1.2.9  Theorems of decidable propositions

Many theorems of logic hold in intuitionistic logic just as they do in classical (non-inuitionistic) logic, for all propositions. Other theorems only hold for decidable propositions, such as the law of the excluded middle (df-dc 805), double negation elimination (notnotrdc 813), or contraposition (condc 823). Our goal is to prove all well-known or important classical theorems, but with suitable decidability conditions so that the proofs follow from intuitionistic axioms. This section is focused on such proofs, given decidability conditions.

Many theorems of this section actually hold for stable propositions (see df-stab 801). Decidable propositions are stable (dcstab 814), but the converse need not hold.

 
Theoremconst 822 Contraposition of a stable proposition. See comment of condc 823. (Contributed by BJ, 18-Nov-2023.)
 |-  (STAB 
 ph  ->  ( ( -.  ph  ->  -.  ps )  ->  ( ps  ->  ph )
 ) )
 
Theoremcondc 823 Contraposition of a decidable proposition.

This theorem swaps or "transposes" the order of the consequents when negation is removed. An informal example is that the statement "if there are no clouds in the sky, it is not raining" implies the statement "if it is raining, there are clouds in the sky." This theorem (without the decidability condition, of course) is called Transp or "the principle of transposition" in Principia Mathematica (Theorem *2.17 of [WhiteheadRussell] p. 103) and is Axiom A3 of [Margaris] p. 49. We will also use the term "contraposition" for this principle, although the reader is advised that in the field of philosophical logic, "contraposition" has a different technical meaning.

(Contributed by Jim Kingdon, 13-Mar-2018.) (Proof shortened by BJ, 18-Nov-2023.)

 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  -.  ps )  ->  ( ps  ->  ph )
 ) )
 
TheoremcondcOLD 824 Obsolete proof of condc 823 as of 18-Nov-2023. (Contributed by Jim Kingdon, 13-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  -.  ps )  ->  ( ps  ->  ph )
 ) )
 
Theorempm2.18dc 825 Proof by contradiction for a decidable proposition. Based on Theorem *2.18 of [WhiteheadRussell] p. 103 (also called Clavius law). Intuitionistically it requires a decidability assumption, but compare with pm2.01 590 which does not. (Contributed by Jim Kingdon, 24-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  ph )  ->  ph )
 )
 
Theoremcon1dc 826 Contraposition for a decidable proposition. Based on theorem *2.15 of [WhiteheadRussell] p. 102. (Contributed by Jim Kingdon, 29-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  ps )  ->  ( -.  ps  ->  ph ) ) )
 
Theoremcon4biddc 827 A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  (DECID 
 ch  ->  ( -.  ps  <->  -.  ch ) ) ) )   =>    |-  ( ph  ->  (DECID  ps  ->  (DECID  ch 
 ->  ( ps  <->  ch ) ) ) )
 
Theoremimpidc 828 An importation inference for a decidable consequent. (Contributed by Jim Kingdon, 30-Apr-2018.)
 |-  (DECID 
 ch  ->  ( ph  ->  ( ps  ->  ch )
 ) )   =>    |-  (DECID 
 ch  ->  ( -.  ( ph  ->  -.  ps )  ->  ch ) )
 
Theoremsimprimdc 829 Simplification given a decidable proposition. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 30-Apr-2018.)
 |-  (DECID 
 ps  ->  ( -.  ( ph  ->  -.  ps )  ->  ps ) )
 
Theoremsimplimdc 830 Simplification for a decidable proposition. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by Jim Kingdon, 29-Mar-2018.)
 |-  (DECID 
 ph  ->  ( -.  ( ph  ->  ps )  ->  ph )
 )
 
Theorempm2.61ddc 831 Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( -.  ps  ->  ch ) )   =>    |-  (DECID 
 ps  ->  ( ph  ->  ch ) )
 
Theorempm2.6dc 832 Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  ps )  ->  (
 ( ph  ->  ps )  ->  ps ) ) )
 
Theoremjadc 833 Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.)
 |-  (DECID 
 ph  ->  ( -.  ph  ->  ch ) )   &    |-  ( ps  ->  ch )   =>    |-  (DECID 
 ph  ->  ( ( ph  ->  ps )  ->  ch )
 )
 
Theoremjaddc 834 Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  th ) ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  (DECID  ps  ->  ( ( ps  ->  ch )  ->  th )
 ) )
 
Theorempm2.61dc 835 Case elimination for a decidable proposition. Theorem *2.61 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( ph  ->  ps )  ->  (
 ( -.  ph  ->  ps )  ->  ps )
 ) )
 
Theorempm2.5gdc 836 Negating an implication for a decidable antecedent. General instance of Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.)
 |-  (DECID 
 ph  ->  ( -.  ( ph  ->  ps )  ->  ( -.  ph  ->  ch )
 ) )
 
Theorempm2.5dc 837 Negating an implication for a decidable antecedent. Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.)
 |-  (DECID 
 ph  ->  ( -.  ( ph  ->  ps )  ->  ( -.  ph  ->  ps )
 ) )
 
Theorempm2.521gdc 838 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107, under a decidability condition. (Contributed by BJ, 28-Oct-2023.)
 |-  (DECID 
 ph  ->  ( -.  ( ph  ->  ps )  ->  ( ch  ->  ph ) ) )
 
Theorempm2.521dc 839 Theorem *2.521 of [WhiteheadRussell] p. 107, but with an additional decidability condition. Note that by replacing in proof pm2.52 630 with conax1k 628, we obtain a proof of the more general instance where the last occurrence of  ph is replaced with any 
ch. (Contributed by Jim Kingdon, 5-May-2018.)
 |-  (DECID 
 ph  ->  ( -.  ( ph  ->  ps )  ->  ( ps  ->  ph ) ) )
 
Theorempm2.521dcALT 840 Alternate proof of pm2.521dc 839. (Contributed by Jim Kingdon, 5-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  (DECID 
 ph  ->  ( -.  ( ph  ->  ps )  ->  ( ps  ->  ph ) ) )
 
Theoremcon34bdc 841 Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.)
 |-  (DECID 
 ps  ->  ( ( ph  ->  ps )  <->  ( -.  ps  ->  -.  ph ) ) )
 
Theoremnotnotbdc 842 Double negation equivalence for a decidable proposition. Like Theorem *4.13 of [WhiteheadRussell] p. 117, but with a decidability antecendent. The forward direction, notnot 603, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 13-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ph  <->  -.  -.  ph )
 )
 
Theoremcon1biimdc 843 Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
 |-  (DECID 
 ph  ->  ( ( -.  ph 
 <->  ps )  ->  ( -.  ps  <->  ph ) ) )
 
Theoremcon1bidc 844 Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( -.  ph 
 <->  ps )  <->  ( -.  ps  <->  ph ) ) ) )
 
Theoremcon2bidc 845 Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  <->  -.  ps )  <->  ( ps  <->  -.  ph ) ) ) )
 
Theoremcon1biddc 846 A contraposition deduction. (Contributed by Jim Kingdon, 4-Apr-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  <->  ch ) ) )   =>    |-  ( ph  ->  (DECID  ps  ->  ( -.  ch  <->  ps ) ) )
 
Theoremcon1biidc 847 A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
 |-  (DECID 
 ph  ->  ( -.  ph  <->  ps ) )   =>    |-  (DECID 
 ph  ->  ( -.  ps  <->  ph ) )
 
Theoremcon1bdc 848 Contraposition. Bidirectional version of con1dc 826. (Contributed by NM, 5-Aug-1993.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( -.  ph  ->  ps )  <->  ( -.  ps  -> 
 ph ) ) ) )
 
Theoremcon2biidc 849 A contraposition inference. (Contributed by Jim Kingdon, 15-Mar-2018.)
 |-  (DECID 
 ps  ->  ( ph  <->  -.  ps ) )   =>    |-  (DECID  ps  ->  ( ps  <->  -.  ph ) )
 
Theoremcon2biddc 850 A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
 |-  ( ph  ->  (DECID  ch  ->  ( ps  <->  -.  ch ) ) )   =>    |-  ( ph  ->  (DECID  ch  ->  ( ch  <->  -.  ps ) ) )
 
Theoremcondandc 851 Proof by contradiction. This only holds for decidable propositions, as it is part of the family of theorems which assume  -.  ps, derive a contradiction, and therefore conclude  ps. By contrast, assuming  ps, deriving a contradiction, and therefore concluding  -.  ps, as in pm2.65 633, is valid for all propositions. (Contributed by Jim Kingdon, 13-May-2018.)
 |-  ( ( ph  /\  -.  ps )  ->  ch )   &    |-  (
 ( ph  /\  -.  ps )  ->  -.  ch )   =>    |-  (DECID  ps  ->  ( ph  ->  ps )
 )
 
Theorembijadc 852 Combine antecedents into a single biconditional. This inference is reminiscent of jadc 833. (Contributed by Jim Kingdon, 4-May-2018.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( -.  ph  ->  ( -.  ps 
 ->  ch ) )   =>    |-  (DECID 
 ps  ->  ( ( ph  <->  ps )  ->  ch ) )
 
Theorempm5.18dc 853 Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider  -.  ( ph  <->  -.  ps ) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  <->  ps ) 
 <->  -.  ( ph  <->  -.  ps ) ) ) )
 
Theoremdfandc 854 Definition of 'and' in terms of negation and implication, for decidable propositions. The forward direction holds for all propositions, and can (basically) be found at pm3.2im 611. (Contributed by Jim Kingdon, 30-Apr-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  /\ 
 ps )  <->  -.  ( ph  ->  -. 
 ps ) ) ) )
 
Theorempm2.13dc 855 A decidable proposition or its triple negation is true. Theorem *2.13 of [WhiteheadRussell] p. 101 with decidability condition added. (Contributed by Jim Kingdon, 13-May-2018.)
 |-  (DECID 
 ph  ->  ( ph  \/  -. 
 -.  -.  ph ) )
 
Theorempm4.63dc 856 Theorem *4.63 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( ph  ->  -.  ps )  <->  (
 ph  /\  ps )
 ) ) )
 
Theorempm4.67dc 857 Theorem *4.67 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 1-May-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( -.  ( -.  ph  ->  -.  ps )  <->  ( -.  ph  /\  ps )
 ) ) )
 
Theoremimanst 858 Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2012.)
 |-  (STAB  ps  ->  ( ( ph  ->  ps )  <->  -.  ( ph  /\  -.  ps ) ) )
 
Theoremimandc 859 Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176, with an added decidability condition. The forward direction, imanim 662, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 25-Apr-2018.)
 |-  (DECID 
 ps  ->  ( ( ph  ->  ps )  <->  -.  ( ph  /\  -.  ps ) ) )
 
Theorempm4.14dc 860 Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)
 |-  (DECID 
 ch  ->  ( ( (
 ph  /\  ps )  ->  ch )  <->  ( ( ph  /\ 
 -.  ch )  ->  -.  ps ) ) )
 
Theorempm2.54dc 861 Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 696, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  ps )  ->  ( ph  \/  ps ) ) )
 
Theoremdfordc 862 Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 696, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( ph  \/  ps )  <->  ( -.  ph  ->  ps ) ) )
 
Theorempm2.25dc 863 Elimination of disjunction based on a disjunction, for a decidable proposition. Based on theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  (DECID 
 ph  ->  ( ph  \/  ( ( ph  \/  ps )  ->  ps )
 ) )
 
Theorempm2.68dc 864 Concluding disjunction from implication for a decidable proposition. Based on theorem *2.68 of [WhiteheadRussell] p. 108. Converse of pm2.62 722 and one half of dfor2dc 865. (Contributed by Jim Kingdon, 27-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( (
 ph  ->  ps )  ->  ps )  ->  ( ph  \/  ps ) ) )
 
Theoremdfor2dc 865 Disjunction expressed in terms of implication only, for a decidable proposition. Based on theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 27-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( ph  \/  ps )  <->  ( ( ph  ->  ps )  ->  ps )
 ) )
 
Theoremimimorbdc 866 Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)
 |-  (DECID 
 ps  ->  ( ( ( ps  ->  ch )  ->  ( ph  ->  ch )
 ) 
 <->  ( ph  ->  ( ps  \/  ch ) ) ) )
 
Theoremimordc 867 Implication in terms of disjunction for a decidable proposition. Based on theorem *4.6 of [WhiteheadRussell] p. 120. The reverse direction, imorr 695, holds for all propositions. (Contributed by Jim Kingdon, 20-Apr-2018.)
 |-  (DECID 
 ph  ->  ( ( ph  ->  ps )  <->  ( -.  ph  \/  ps ) ) )
 
Theorempm4.62dc 868 Implication in terms of disjunction. Like Theorem *4.62 of [WhiteheadRussell] p. 120, but for a decidable antecedent. (Contributed by Jim Kingdon, 21-Apr-2018.)
 |-  (DECID 
 ph  ->  ( ( ph  ->  -.  ps )  <->  ( -.  ph  \/  -.  ps ) ) )
 
Theoremianordc 869 Negated conjunction in terms of disjunction (DeMorgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120, but where one proposition is decidable. The reverse direction, pm3.14 727, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.)
 |-  (DECID 
 ph  ->  ( -.  ( ph  /\  ps )  <->  ( -.  ph  \/  -.  ps ) ) )
 
Theorempm4.64dc 870 Theorem *4.64 of [WhiteheadRussell] p. 120, given a decidability condition. The reverse direction, pm2.53 696, holds for all propositions. (Contributed by Jim Kingdon, 2-May-2018.)
 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  ps )  <->  ( ph  \/  ps ) ) )
 
Theorempm4.66dc 871 Theorem *4.66 of [WhiteheadRussell] p. 120, given a decidability condition. (Contributed by Jim Kingdon, 2-May-2018.)
 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  -.  ps )  <->  (
 ph  \/  -.  ps )
 ) )
 
Theorempm4.54dc 872 Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( -.  ph  /\  ps )  <->  -.  ( ph  \/  -. 
 ps ) ) ) )
 
Theorempm4.79dc 873 Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ( ps  ->  ph )  \/  ( ch  ->  ph )
 ) 
 <->  ( ( ps  /\  ch )  ->  ph ) ) ) )
 
Theorempm5.17dc 874 Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)
 |-  (DECID 
 ps  ->  ( ( (
 ph  \/  ps )  /\  -.  ( ph  /\  ps ) )  <->  ( ph  <->  -.  ps ) ) )
 
Theorempm2.85dc 875 Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.)
 |-  (DECID 
 ph  ->  ( ( (
 ph  \/  ps )  ->  ( ph  \/  ch ) )  ->  ( ph  \/  ( ps  ->  ch )
 ) ) )
 
Theoremorimdidc 876 Disjunction distributes over implication. The forward direction, pm2.76 782, is valid intuitionistically. The reverse direction holds if  ph is decidable, as can be seen at pm2.85dc 875. (Contributed by Jim Kingdon, 1-Apr-2018.)
 |-  (DECID 
 ph  ->  ( ( ph  \/  ( ps  ->  ch )
 ) 
 <->  ( ( ph  \/  ps )  ->  ( ph  \/  ch ) ) ) )
 
Theorempm2.26dc 877 Decidable proposition version of theorem *2.26 of [WhiteheadRussell] p. 104. (Contributed by Jim Kingdon, 20-Apr-2018.)
 |-  (DECID 
 ph  ->  ( -.  ph  \/  ( ( ph  ->  ps )  ->  ps )
 ) )
 
Theorempm4.81dc 878 Theorem *4.81 of [WhiteheadRussell] p. 122, for decidable propositions. This one needs a decidability condition, but compare with pm4.8 681 which holds for all propositions. (Contributed by Jim Kingdon, 4-Jul-2018.)
 |-  (DECID 
 ph  ->  ( ( -.  ph  ->  ph )  <->  ph ) )
 
Theorempm5.11dc 879 A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.)
 |-  (DECID 
 ph  ->  (DECID 
 ps  ->  ( ( ph  ->  ps )  \/  ( -.  ph  ->  ps )
 ) ) )
 
Theorempm5.12dc 880 Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
 |-  (DECID 
 ps  ->  ( ( ph  ->  ps )  \/  ( ph  ->  -.  ps )
 ) )
 
Theorempm5.14dc 881 A decidable proposition is implied by or implies other propositions. Based on theorem *5.14 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
 |-  (DECID 
 ps  ->  ( ( ph  ->  ps )  \/  ( ps  ->  ch ) ) )
 
Theorempm5.13dc 882 An implication holds in at least one direction, where one proposition is decidable. Based on theorem *5.13 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)
 |-  (DECID 
 ps  ->  ( ( ph  ->  ps )  \/  ( ps  ->  ph ) ) )
 
Theorempm5.55dc 883 A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( (
 ph  \/  ps )  <->  ph )  \/  ( (
 ph  \/  ps )  <->  ps ) ) )
 
Theorempeircedc 884 Peirce's theorem for a decidable proposition. This odd-looking theorem can be seen as an alternative to exmiddc 806, condc 823, or notnotrdc 813 in the sense of expressing the "difference" between an intuitionistic system of propositional calculus and a classical system. In intuitionistic logic, it only holds for decidable propositions. (Contributed by Jim Kingdon, 3-Jul-2018.)
 |-  (DECID 
 ph  ->  ( ( (
 ph  ->  ps )  ->  ph )  -> 
 ph ) )
 
Theoremlooinvdc 885 The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 865, we can see that this expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability condition). (Contributed by NM, 12-Aug-2004.)
 |-  (DECID 
 ph  ->  ( ( (
 ph  ->  ps )  ->  ps )  ->  ( ( ps  ->  ph )  ->  ph ) ) )
 
1.2.10  Miscellaneous theorems of propositional calculus
 
Theorempm5.21nd 886 Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.)
 |-  ( ( ph  /\  ps )  ->  th )   &    |-  ( ( ph  /\ 
 ch )  ->  th )   &    |-  ( th  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorempm5.35 887 Theorem *5.35 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  <->  ch ) ) )
 
Theorempm5.54dc 888 A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)
 |-  (DECID 
 ph  ->  ( ( (
 ph  /\  ps )  <->  ph )  \/  ( (
 ph  /\  ps )  <->  ps ) ) )
 
Theorembaib 889 Move conjunction outside of biconditional. (Contributed by NM, 13-May-1999.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ph 
 <->  ch ) )
 
Theorembaibr 890 Move conjunction outside of biconditional. (Contributed by NM, 11-Jul-1994.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ps  ->  ( ch 
 <-> 
 ph ) )
 
Theoremrbaib 891 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ch  ->  ( ph 
 <->  ps ) )
 
Theoremrbaibr 892 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  <->  ( ps  /\  ch ) )   =>    |-  ( ch  ->  ( ps 
 <-> 
 ph ) )
 
Theorembaibd 893 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 ch )  ->  ( ps 
 <-> 
 th ) )
 
Theoremrbaibd 894 Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.)
 |-  ( ph  ->  ( ps 
 <->  ( ch  /\  th ) ) )   =>    |-  ( ( ph  /\ 
 th )  ->  ( ps 
 <->  ch ) )
 
Theorempm5.44 895 Theorem *5.44 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  ->  ch )  <->  ( ph  ->  ( ps  /\  ch )
 ) ) )
 
Theorempm5.6dc 896 Conjunction in antecedent versus disjunction in consequent, for a decidable proposition. Theorem *5.6 of [WhiteheadRussell] p. 125, with decidability condition added. The reverse implication holds for all propositions (see pm5.6r 897). (Contributed by Jim Kingdon, 2-Apr-2018.)
 |-  (DECID 
 ps  ->  ( ( (
 ph  /\  -.  ps )  ->  ch )  <->  ( ph  ->  ( ps  \/  ch )
 ) ) )
 
Theorempm5.6r 897 Conjunction in antecedent versus disjunction in consequent. One direction of Theorem *5.6 of [WhiteheadRussell] p. 125. If  ps is decidable, the converse also holds (see pm5.6dc 896). (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( ( ph  ->  ( ps  \/  ch )
 )  ->  ( ( ph  /\  -.  ps )  ->  ch ) )
 
Theoremorcanai 898 Change disjunction in consequent to conjunction in antecedent. (Contributed by NM, 8-Jun-1994.)
 |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ( ph  /\  -.  ps )  ->  ch )
 
Theoremintnan 899 Introduction of conjunct inside of a contradiction. (Contributed by NM, 16-Sep-1993.)
 |- 
 -.  ph   =>    |- 
 -.  ( ps  /\  ph )
 
Theoremintnanr 900 Introduction of conjunct inside of a contradiction. (Contributed by NM, 3-Apr-1995.)
 |- 
 -.  ph   =>    |- 
 -.  ( ph  /\  ps )
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