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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | impbida 801 | Deduce an equivalence from two implications. Variant of impbid 212. (Contributed by NM, 17-Feb-2007.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜒) → 𝜓) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | ||
Theorem | pm5.21nd 802 | Eliminate an antecedent implied by each side of a biconditional. Variant of pm5.21ndd 379. (Contributed by NM, 20-Nov-2005.) (Proof shortened by Wolf Lammen, 4-Nov-2013.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜃) & ⊢ ((𝜑 ∧ 𝜒) → 𝜃) & ⊢ (𝜃 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | ||
Theorem | pm3.35 803 | Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. Variant of pm2.27 42. (Contributed by NM, 14-Dec-2002.) |
⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓) | ||
Theorem | pm5.74da 804 | Distribution of implication over biconditional (deduction form). Variant of pm5.74d 273. (Contributed by NM, 4-May-2007.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃))) | ||
Theorem | bitr 805 | Theorem *4.22 of [WhiteheadRussell] p. 117. bitri 275 in closed form. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜓 ↔ 𝜒)) → (𝜑 ↔ 𝜒)) | ||
Theorem | biantr 806 | A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.) |
⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜓)) → (𝜑 ↔ 𝜒)) | ||
Theorem | pm4.14 807 | Theorem *4.14 of [WhiteheadRussell] p. 117. Related to con34b 316. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.) |
⊢ (((𝜑 ∧ 𝜓) → 𝜒) ↔ ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)) | ||
Theorem | pm3.37 808 | Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.) |
⊢ (((𝜑 ∧ 𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓)) | ||
Theorem | anim12 809 | Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 609. Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it praeclarum theorema (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))) | ||
Theorem | pm3.4 810 | Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | exbiri 811 | Inference form of exbir 44475. This proof is exbiriVD 44851 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 812 | Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((¬ 𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜓 → 𝜒) | ||
Theorem | pm2.61dan 813 | Elimination of an antecedent. (Contributed by NM, 1-Jan-2005.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | pm2.61ddan 814 | 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 815 | Elimination of two antecedents. (Contributed by NM, 9-Jul-2013.) |
⊢ ((𝜑 ∧ ¬ 𝜓) → 𝜃) & ⊢ ((𝜑 ∧ ¬ 𝜒) → 𝜃) & ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | mtand 816 | A modus tollens deduction. (Contributed by Jeff Hankins, 19-Aug-2009.) |
⊢ (𝜑 → ¬ 𝜒) & ⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
Theorem | pm2.65da 817 | Deduction for proof by contradiction. (Contributed by NM, 12-Jun-2014.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝜒) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
Theorem | condan 818 | 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 819 | 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 822 and elelb 36879. (Contributed by BJ, 4-Mar-2023.) (Proof shortened by Wolf Lammen, 8-Mar-2023.) |
⊢ ((𝜑 → 𝜓) ↔ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒)))) | ||
Theorem | biadani 820 | Inference associated with biadan 819. (Contributed by BJ, 4-Mar-2023.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | biadaniALT 821 | Alternate proof of biadani 820 not using biadan 819. (Contributed by BJ, 4-Mar-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))) | ||
Theorem | biadanii 822 | Inference associated with biadani 820. 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 | biadanid 823 | Deduction associated with biadani 820. Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) & ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) ⇒ ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) | ||
Theorem | pm5.1 824 | 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 | pm5.21 825 | 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 | pm5.35 826 | Theorem *5.35 of [WhiteheadRussell] p. 125. Closed form of 2thd 265. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ↔ 𝜒))) | ||
Theorem | abai 827 | 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 828 | Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.) |
⊢ (𝜑 ↔ (𝜑 ∧ (𝜓 → 𝜑))) | ||
Theorem | impimprbi 829 | An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 474, but ↔ is a weaker operator than ∧. Note that an implication and its reverse can never be simultaneously false, because of pm2.521 176. (Contributed by Wolf Lammen, 18-Dec-2023.) |
⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ↔ (𝜓 → 𝜑))) | ||
Theorem | nan 830 | Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.) |
⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒)) | ||
Theorem | pm5.31 831 | Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜓 ∧ 𝜒))) | ||
Theorem | pm5.31r 832 | Variant of pm5.31 831. (Contributed by Rodolfo Medina, 15-Oct-2010.) |
⊢ ((𝜒 ∧ (𝜑 → 𝜓)) → (𝜑 → (𝜒 ∧ 𝜓))) | ||
Theorem | pm4.15 833 | 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 | pm5.36 834 | Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∧ (𝜑 ↔ 𝜓)) ↔ (𝜓 ∧ (𝜑 ↔ 𝜓))) | ||
Theorem | annotanannot 835 | 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 | pm5.33 836 | Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∧ (𝜓 → 𝜒)) ↔ (𝜑 ∧ ((𝜑 ∧ 𝜓) → 𝜒))) | ||
Theorem | syl12anc 837 | 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 838 | 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 839 | 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 840 | Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1373 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 | syldbl2 841 | Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.) |
⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃)) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜃) | ||
Theorem | mpsyl4anc 842 | An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.) |
⊢ 𝜑 & ⊢ 𝜓 & ⊢ 𝜒 & ⊢ (𝜃 → 𝜏) & ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜏) → 𝜂) ⇒ ⊢ (𝜃 → 𝜂) | ||
Theorem | pm4.87 843 | 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 844 | Removal of conjunct from one side of an equivalence. (Contributed by NM, 21-Jun-1993.) |
⊢ (((𝜑 → 𝜓) ∧ (𝜒 ↔ (𝜓 ∧ 𝜑))) → (𝜒 ↔ 𝜑)) | ||
Theorem | a2and 845 | 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 846 | 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 396 (theorems that are proved using df-an 396 are deferred to the next section). Basic theorems that help simplifying and applying disjunction are olc 868, orc 867, and orcom 870. 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 984: 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 406 and ex 412 as noted in the previous section. An illustration of the conservativity of df-an 396 is given by orim12dALT 911, which is an alternate proof of orim12d 966 not using df-an 396. | ||
Syntax | wo 847 | Extend wff definition to include disjunction ("or"). |
wff (𝜑 ∨ 𝜓) | ||
Definition | df-or 848 |
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 30449). After we define the constant
true ⊤ (df-tru 1539) and the constant false ⊥ (df-fal 1549), we
will be able to prove these truth table values:
((⊤ ∨ ⊤) ↔ ⊤) (truortru 1573), ((⊤ ∨ ⊥)
↔ ⊤)
(truorfal 1574), ((⊥ ∨ ⊤)
↔ ⊤) (falortru 1575), and
((⊥ ∨ ⊥) ↔ ⊥) (falorfal 1576).
Contrast with ∧ (df-an 396), → (wi 4), ⊼ (df-nan 1488), and ⊻ (df-xor 1508). (Contributed by NM, 27-Dec-1992.) |
⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓)) | ||
Theorem | pm4.64 849 | Theorem *4.64 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 → 𝜓) ↔ (𝜑 ∨ 𝜓)) | ||
Theorem | pm4.66 850 | Theorem *4.66 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm2.53 851 | Theorem *2.53 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) | ||
Theorem | pm2.54 852 | Theorem *2.54 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)) | ||
Theorem | imor 853 | Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) |
⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓)) | ||
Theorem | imori 854 | Infer disjunction from implication. (Contributed by NM, 12-Mar-2012.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (¬ 𝜑 ∨ 𝜓) | ||
Theorem | imorri 855 | Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (¬ 𝜑 ∨ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | pm4.62 856 | Theorem *4.62 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | jaoi 857 | Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) |
⊢ (𝜑 → 𝜓) & ⊢ (𝜒 → 𝜓) ⇒ ⊢ ((𝜑 ∨ 𝜒) → 𝜓) | ||
Theorem | jao1i 858 | 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 859 | 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 860 | 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 861 | Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) |
⊢ (𝜑 ∨ 𝜓) ⇒ ⊢ (¬ 𝜑 → 𝜓) | ||
Theorem | orri 862 | Infer disjunction from implication. (Contributed by NM, 11-Jun-1994.) |
⊢ (¬ 𝜑 → 𝜓) ⇒ ⊢ (𝜑 ∨ 𝜓) | ||
Theorem | orrd 863 | Deduce disjunction from implication. (Contributed by NM, 27-Nov-1995.) |
⊢ (𝜑 → (¬ 𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) | ||
Theorem | ord 864 | Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → (¬ 𝜓 → 𝜒)) | ||
Theorem | orci 865 | Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
⊢ 𝜑 ⇒ ⊢ (𝜑 ∨ 𝜓) | ||
Theorem | olci 866 | Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Proof shortened by Wolf Lammen, 14-Nov-2012.) |
⊢ 𝜑 ⇒ ⊢ (𝜓 ∨ 𝜑) | ||
Theorem | orc 867 | Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) |
⊢ (𝜑 → (𝜑 ∨ 𝜓)) | ||
Theorem | olc 868 | Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) |
⊢ (𝜑 → (𝜓 ∨ 𝜑)) | ||
Theorem | pm1.4 869 | Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 ∨ 𝜓) → (𝜓 ∨ 𝜑)) | ||
Theorem | orcom 870 | 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 871 | Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → (𝜒 ∨ 𝜓)) | ||
Theorem | orcoms 872 | Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.) |
⊢ ((𝜑 ∨ 𝜓) → 𝜒) ⇒ ⊢ ((𝜓 ∨ 𝜑) → 𝜒) | ||
Theorem | orcd 873 | Deduction introducing a disjunct. A translation of natural deduction rule ∨ IR (∨ insertion right), see natded 30431. (Contributed by NM, 20-Sep-2007.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) | ||
Theorem | olcd 874 | Deduction introducing a disjunct. A translation of natural deduction rule ∨ IL (∨ insertion left), see natded 30431. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → (𝜒 ∨ 𝜓)) | ||
Theorem | orcs 875 | 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 876 | Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.) |
⊢ ((𝜑 ∨ 𝜓) → 𝜒) ⇒ ⊢ (𝜓 → 𝜒) | ||
Theorem | olcnd 877 | A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) (Proof shortened by Wolf Lammen, 13-Apr-2024.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → ¬ 𝜒) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | orcnd 878 | A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → 𝜒) | ||
Theorem | mtord 879 | 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 880 | Infer negated disjunction of negated premises. (Contributed by NM, 4-Apr-1995.) |
⊢ ¬ 𝜑 & ⊢ ¬ 𝜓 ⇒ ⊢ ¬ (𝜑 ∨ 𝜓) | ||
Theorem | pm2.45 881 | Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑) | ||
Theorem | pm2.46 882 | Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓) | ||
Theorem | pm2.47 883 | Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ 𝜓)) | ||
Theorem | pm2.48 884 | Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ∨ ¬ 𝜓)) | ||
Theorem | pm2.49 885 | Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)) | ||
Theorem | norbi 886 | If neither of two propositions is true, then these propositions are equivalent. (Contributed by BJ, 26-Apr-2019.) |
⊢ (¬ (𝜑 ∨ 𝜓) → (𝜑 ↔ 𝜓)) | ||
Theorem | nbior 887 | 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 888 | 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 | pm2.25 889 | Theorem *2.25 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ∨ ((𝜑 ∨ 𝜓) → 𝜓)) | ||
Theorem | orel2 890 | 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 | pm2.67-2 891 | Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | pm2.67 892 | Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 ∨ 𝜓) → 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | curryax 893 | A non-intuitionistic positive statement, sometimes called a paradox of material implication. Sometimes called Curry's axiom. Similar to exmid 894 (obtained by substituting ⊥ for 𝜓) but positive. For another non-intuitionistic positive statement, see peirce 202. (Contributed by BJ, 4-Apr-2021.) |
⊢ (𝜑 ∨ (𝜑 → 𝜓)) | ||
Theorem | exmid 894 | 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 895 | Law of excluded middle in a context. (Contributed by Mario Carneiro, 9-Feb-2017.) |
⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓)) | ||
Theorem | pm2.1 896 | Theorem *2.1 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Nov-2012.) |
⊢ (¬ 𝜑 ∨ 𝜑) | ||
Theorem | pm2.13 897 | Theorem *2.13 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.) |
⊢ (𝜑 ∨ ¬ ¬ ¬ 𝜑) | ||
Theorem | pm2.621 898 | Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) |
⊢ ((𝜑 → 𝜓) → ((𝜑 ∨ 𝜓) → 𝜓)) | ||
Theorem | pm2.62 899 | 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 | pm2.68 900 | Theorem *2.68 of [WhiteheadRussell] p. 108. (Contributed by NM, 3-Jan-2005.) |
⊢ (((𝜑 → 𝜓) → 𝜓) → (𝜑 ∨ 𝜓)) |
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