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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | exbiri 801 | Inference form of exbir 39638. This proof is exbiriVD 40023 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof shortened by Wolf Lammen, 27-Jan-2013.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒))) | ||
Theorem | pm2.61ian 802 | Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜓 → 𝜒) | ||
Theorem | pm2.61dan 803 | Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | pm2.61ddan 804 | Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜃) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) & ⊢ ((𝜑 ∧ (¬ 𝜓 ∧ ¬ 𝜒)) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | pm2.61dda 805 | Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃) & ⊢ ((𝜑 ∧ ¬ 𝜒) → 𝜃) & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | mtand 806 | A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
Theorem | pm2.65da 807 | Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
Theorem | condan 808 | Proof by contradiction. (Contributed by NM, 9-Feb-2006.) (Proof shortened by Wolf Lammen, 19-Jun-2014.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ 𝜒) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | biadan 809 | An implication is equivalent to the equivalence of some implied equivalence and some other equivalence involving a conjunction. A utility lemma as illustrated in biadanii 813 and elelb 33456. (Contributed by BJ, 4-Mar-2023.) (Proof shortened by Wolf Lammen, 8-Mar-2023.) |
⊢ ((𝜑 → 𝜓) ↔ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))) | ||
Theorem | biadanOLD 810 | Obsolete version of biadan 809 as of 8-Mar-2023. (Contributed by BJ, 4-Mar-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝜑 → 𝜓) ↔ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))) | ||
Theorem | biadani 811 | Inference associated with biadan 809. (Contributed by BJ, 4-Mar-2023.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | biadaniALT 812 | Alternate proof of biadani 811 not using biadan 809. (Contributed by BJ, 4-Mar-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | biadanii 813 | Inference associated with biadani 811. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) | ||
Theorem | biadan2OLD 814 | Obsolete proof of biadanii 813 as of 4-Mar-2023. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜓 → (𝜑 ↔ 𝜒)) ⇒ ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) | ||
Theorem | pm5.1 815 | Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ 𝜓)) | ||
Theorem | pm5.21 816 | Two propositions are equivalent if they are both false. Theorem *5.21 of [WhiteheadRussell] p. 124. (Contributed by NM, 21-May-1994.) |
⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑 ↔ 𝜓)) | ||
Theorem | pm5.35 817 | Theorem *5.35 of [WhiteheadRussell] p. 125. Closed form of 2thd 257. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) | ||
Theorem | abai 818 | Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.) |
⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓))) | ||
Theorem | pm4.45im 819 | Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.) |
⊢ (𝜑 ↔ (𝜑 ∧ (𝜓 → 𝜑))) | ||
Theorem | nan 820 | Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.) |
⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒)) | ||
Theorem | pm5.31 821 | Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜓 ∧ 𝜒))) | ||
Theorem | pm5.31r 822 | Variant of pm5.31 821. (Contributed by Rodolfo Medina, 15-Oct-2010.) |
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓))) | ||
Theorem | pm4.15 823 | Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.) |
⊢ (((𝜑 ∧ 𝜓) → ¬ 𝜒) ↔ ((𝜓 ∧ 𝜒) → ¬ 𝜑)) | ||
Theorem | pm5.36 824 | Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) ↔ (𝜓 ∧ (𝜑 ↔ 𝜓))) | ||
Theorem | annotanannot 825 | A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022.) (Proof shortened by Wolf Lammen, 1-Apr-2022.) |
⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ ¬ 𝜓)) | ||
Theorem | pm5.33 826 | Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒))) | ||
Theorem | syl12anc 827 | Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃)) → 𝜏) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | syl21anc 828 | Syllogism combined with contraction. (Contributed by Jeff Hankins, 1-Aug-2009.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ (𝜑 → 𝜏) | ||
Theorem | syl22anc 829 | Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → 𝜏) & ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏)) → 𝜂) ⇒ ⊢ (𝜑 → 𝜂) | ||
Theorem | syl1111anc 830 | Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1442 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜑 → 𝜒) & ⊢ (𝜑 → 𝜃) & ⊢ (𝜑 → 𝜏) & ⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜑 → 𝜂) | ||
Theorem | mpsyl4anc 831 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ (𝜃 → 𝜏) & ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜃 → 𝜂) | ||
Theorem | pm4.87 832 | Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.) |
⊢ (((((𝜑 ∧ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 → 𝜒))) ∧ ((𝜑 → (𝜓 → 𝜒)) ↔ (𝜓 → (𝜑 → 𝜒)))) ∧ ((𝜓 → (𝜑 → 𝜒)) ↔ ((𝜓 ∧ 𝜑) → 𝜒))) | ||
Theorem | bimsc1 833 | Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑)) | ||
Theorem | a2and 834 | Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) |
⊢ (𝜑 → ((𝜓 ∧ 𝜌) → (𝜏 → 𝜃))) & ⊢ (𝜑 → ((𝜓 ∧ 𝜌) → 𝜒)) ⇒ ⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜏) → ((𝜓 ∧ 𝜌) → 𝜃))) | ||
Theorem | animpimp2impd 835 | Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.) |
⊢ ((𝜓 ∧ 𝜑) → (𝜒 → (𝜃 → 𝜂))) & ⊢ ((𝜓 ∧ (𝜑 ∧ 𝜃)) → (𝜂 → 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → (𝜃 → 𝜏)))) | ||
This section defines disjunction of two formulas, denoted by infix "∨ " and read "or". It is defined in terms of implication and negation, which is possible in classical logic (but not in intuitionistic logic: see iset.mm). This section contains only theorems proved without df-an 387 (theorems that are proved using df-an 387 are deferred to the next section). Basic theorems that help simplifying and applying disjunction are olc 857, orc 856, and orcom 859. As mentioned in the "note on definitions" in the section comment for logical equivalence, all theorems in this and the previous section can be stated in terms of implication and negation only. Additionally, in classical logic (but not in intuitionistic logic: see iset.mm), it is also possible to translate conjunction into disjunction and conversely via the De Morgan law anor 968: conjunction and disjunction are dual connectives. Either is sufficient to develop all propositional calculus of the logic (together with implication and negation). In practice, conjunction is more efficient, its big advantage being the possibility to use it to group antecedents in a convenient way, using imp 397 and ex 403 as noted in the previous section. An illustration of the conservativity of df-an 387 is given by orim12dALT 898, which is an alternate proof of orim12d 950 not using df-an 387. | ||
Syntax | wo 836 | Extend wff definition to include disjunction ("or"). |
wff (𝜑 ∨ 𝜓) | ||
Definition | df-or 837 |
Define disjunction (logical "or"). Definition of [Margaris] p. 49. When
the left operand, right operand, or both are true, the result is true;
when both sides are false, the result is false. For example, it is true
that (2 = 3 ∨ 4 = 4) (ex-or 27853). After we define the constant
true ⊤ (df-tru 1605) and the constant false ⊥ (df-fal 1615), we
will be able to prove these truth table values:
((⊤ ∨ ⊤) ↔ ⊤) (truortru 1639), ((⊤ ∨ ⊥)
↔ ⊤)
(truorfal 1640), ((⊥ ∨ ⊤)
↔ ⊤) (falortru 1641), and
((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1642).
Contrast with ∧ (df-an 387), → (wi 4), ⊼ (df-nan 1558), and ⊻ (df-xor 1583). (Contributed by NM, 27-Dec-1992.) |
⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | ||
Theorem | pm4.64 838 | Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓)) | ||
Theorem | pm4.66 839 | Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm2.53 840 | Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) | ||
Theorem | pm2.54 841 | Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)) | ||
Theorem | imor 842 | Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) |
⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | ||
Theorem | imori 843 | Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (¬ 𝜑 ∨ 𝜓) | ||
Theorem | imorri 844 | Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (¬ 𝜑 ∨ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | pm4.62 845 | Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | jaoi 846 | Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) → 𝜓) | ||
Theorem | jao1i 847 | Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.) |
⊢ (𝜓 → (𝜒 → 𝜑)) ⇒ ⊢ ((𝜑 ∨ 𝜓) → (𝜒 → 𝜑)) | ||
Theorem | jaod 848 | Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜒)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → 𝜒)) | ||
Theorem | mpjaod 849 | Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜒)) & ⊢ (𝜑 → (𝜓 ∨ 𝜃)) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | ori 850 | Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) |
⊢ (𝜑 ∨ 𝜓) ⇒ ⊢ (¬ 𝜑 → 𝜓) | ||
Theorem | orri 851 | Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.) |
⊢ (¬ 𝜑 → 𝜓) ⇒ ⊢ (𝜑 ∨ 𝜓) | ||
Theorem | orrd 852 | Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.) |
⊢ (𝜑 → (¬ 𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) | ||
Theorem | ord 853 | Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → (¬ 𝜓 → 𝜒)) | ||
Theorem | orci 854 | Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
⊢ 𝜑 ⇒ ⊢ (𝜑 ∨ 𝜓) | ||
Theorem | olci 855 | Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
⊢ 𝜑 ⇒ ⊢ (𝜓 ∨ 𝜑) | ||
Theorem | orc 856 | Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) |
⊢ (𝜑 → (𝜑 ∨ 𝜓)) | ||
Theorem | olc 857 | Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) |
⊢ (𝜑 → (𝜓 ∨ 𝜑)) | ||
Theorem | pm1.4 858 | Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜓) → (𝜓 ∨ 𝜑)) | ||
Theorem | orcom 859 | Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.) |
⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | ||
Theorem | orcomd 860 | Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → (𝜒 ∨ 𝜓)) | ||
Theorem | orcoms 861 | Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.) |
⊢ ((𝜑 ∨ 𝜓) → 𝜒) ⇒ ⊢ ((𝜓 ∨ 𝜑) → 𝜒) | ||
Theorem | orcd 862 | Deduction introducing a disjunct. A translation of natural deduction rule ∨ IR (∨ insertion right), see natded 27835. (Contributed by NM, 20-Sep-2007.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) | ||
Theorem | olcd 863 | Deduction introducing a disjunct. A translation of natural deduction rule ∨ IL (∨ insertion left), see natded 27835. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 ∨ 𝜓)) | ||
Theorem | orcs 864 | Deduction eliminating disjunct. Notational convention: We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 17) -type inference in a proof. (Contributed by NM, 21-Jun-1994.) |
⊢ ((𝜑 ∨ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | olcs 865 | Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
⊢ ((𝜑 ∨ 𝜓) → 𝜒) ⇒ ⊢ (𝜓 → 𝜒) | ||
Theorem | mtord 866 | A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ (𝜑 → ¬ 𝜃) & ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
Theorem | pm3.2ni 867 | Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) |
⊢ ¬ 𝜑 & ⊢ ¬ 𝜓 ⇒ ⊢ ¬ (𝜑 ∨ 𝜓) | ||
Theorem | pm2.45 868 | Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑) | ||
Theorem | pm2.46 869 | Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓) | ||
Theorem | pm2.47 870 | Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ 𝜓)) | ||
Theorem | pm2.48 871 | Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm2.49 872 | Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | norbi 873 | If neither of two propositions is true, then these propositions are equivalent. (Contributed by BJ, 26-Apr-2019.) |
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) | ||
Theorem | nbior 874 | If two propositions are not equivalent, then at least one is true. (Contributed by BJ, 19-Apr-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.) |
⊢ (¬ (𝜑 ↔ 𝜓) → (𝜑 ∨ 𝜓)) | ||
Theorem | orel1 875 | Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 21-Jul-2012.) |
⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) → 𝜓)) | ||
Theorem | pm2.25 876 | Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓)) | ||
Theorem | orel2 877 | Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Apr-2013.) |
⊢ (¬ 𝜑 → ((𝜓 ∨ 𝜑) → 𝜓)) | ||
Theorem | pm2.67-2 878 | Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | pm2.67 879 | Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) → 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | curryax 880 | A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. Similar to exmid 881 but positive. For another non-intuitionistic positive statement, see peirce 194. (Contributed by BJ, 4-Apr-2021.) |
⊢ (𝜑 ∨ (𝜑 → 𝜓)) | ||
Theorem | exmid 881 | Law of excluded middle, 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. This is an essential distinction of our classical logic and is not a theorem of intuitionistic logic. In intuitionistic logic, if this statement is true for some 𝜑, then 𝜑 is decidable. (Contributed by NM, 29-Dec-1992.) |
⊢ (𝜑 ∨ ¬ 𝜑) | ||
Theorem | exmidd 882 | Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) | ||
Theorem | pm2.1 883 | Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) |
⊢ (¬ 𝜑 ∨ 𝜑) | ||
Theorem | pm2.13 884 | Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ∨ ¬ ¬ ¬ 𝜑) | ||
Theorem | pm2.621 885 | Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) → ((𝜑 ∨ 𝜓) → 𝜓)) | ||
Theorem | pm2.62 886 | Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.) |
⊢ ((𝜑 ∨ 𝜓) → ((𝜑 → 𝜓) → 𝜓)) | ||
Theorem | pm2.68 887 | Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)) | ||
Theorem | dfor2 888 | Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Wolf Lammen, 20-Oct-2012.) |
⊢ ((𝜑 ∨ 𝜓) ↔ ((𝜑 → 𝜓) → 𝜓)) | ||
Theorem | pm2.07 889 | Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 → (𝜑 ∨ 𝜑)) | ||
Theorem | pm1.2 890 | Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜑) → 𝜑) | ||
Theorem | oridm 891 | Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.) |
⊢ ((𝜑 ∨ 𝜑) ↔ 𝜑) | ||
Theorem | pm4.25 892 | Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ↔ (𝜑 ∨ 𝜑)) | ||
Theorem | pm2.4 893 | Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓)) | ||
Theorem | pm2.41 894 | Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜓 ∨ (𝜑 ∨ 𝜓)) → (𝜑 ∨ 𝜓)) | ||
Theorem | orim12i 895 | Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜃) ⇒ ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃)) | ||
Theorem | orim1i 896 | Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜒)) | ||
Theorem | orim2i 897 | Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜒 ∨ 𝜑) → (𝜒 ∨ 𝜓)) | ||
Theorem | orim12dALT 898 | Alternate proof of orim12d 950 which does not depend on df-an 387. This is an illustration of the conservativity of definitions (definitions do not permit to prove additional theorems whose statements do not contain the defined symbol). (Contributed by Wolf Lammen, 8-Aug-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜃 → 𝜏)) ⇒ ⊢ (𝜑 → ((𝜓 ∨ 𝜃) → (𝜒 ∨ 𝜏))) | ||
Theorem | orbi2i 899 | Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓)) | ||
Theorem | orbi1i 900 | Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜓 ∨ 𝜒)) |
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