P. 8 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 701-800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2false 701	Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- -. ph   &    |- -. ps   =>    |- ( ph <-> ps )
Theorem	2falsed 702	Two falsehoods are equivalent (deduction form). (Contributed by NM, 11-Oct-2013.)
|- ( ph -> -. ps )   &    |- ( ph -> -. ch )   =>    |- ( ph -> ( ps <-> ch ) )
Theorem	pm5.21ni 703	Two propositions implying a false one are equivalent. (Contributed by NM, 16-Feb-1996.) (Proof shortened by Wolf Lammen, 19-May-2013.)
|- ( ph -> ps )   &    |- ( ch -> ps )   =>    |- ( -. ps -> ( ph <-> ch ) )
Theorem	pm5.21nii 704	Eliminate an antecedent implied by each side of a biconditional. (Contributed by NM, 21-May-1999.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ph -> ps )   &    |- ( ch -> ps )   &    |- ( ps -> ( ph <-> ch ) )   =>    |- ( ph <-> ch )
Theorem	pm5.21ndd 705	Eliminate an antecedent implied by each side of a biconditional, deduction version. (Contributed by Paul Chapman, 21-Nov-2012.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ph -> ( ch -> ps ) )   &    |- ( ph -> ( th -> ps ) )   &    |- ( ph -> ( ps -> ( ch <-> th ) ) )   =>    |- ( ph -> ( ch <-> th ) )
Theorem	pm5.19 706	Theorem *5.19 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- -. ( ph <-> -. ph )
Theorem	pm4.8 707	Theorem *4.8 of [WhiteheadRussell] p. 122. This one holds for all propositions, but compare with pm4.81dc 908 which requires a decidability condition. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph -> -. ph ) <-> -. ph )
1.2.6  Logical disjunction
Syntax	wo 708	Extend wff definition to include disjunction ('or').
wff ( ph \/ ps )
Axiom	ax-io 709	Definition of 'or'. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias jaob 710 instead. (New usage is discouraged.)
|- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) )
Theorem	jaob 710	Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. Alias of ax-io 709. (Contributed by NM, 30-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) )
Theorem	olc 711	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
|- ( ph -> ( ps \/ ph ) )
Theorem	orc 712	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104. (Contributed by NM, 30-Aug-1993.) (Revised by NM, 31-Jan-2015.)
|- ( ph -> ( ph \/ ps ) )
Theorem	pm2.67-2 713	Slight generalization of Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
|- ( ( ( ph \/ ch ) -> ps ) -> ( ph -> ps ) )
Theorem	oibabs 714	Absorption of disjunction into equivalence. (Contributed by NM, 6-Aug-1995.) (Proof shortened by Wolf Lammen, 3-Nov-2013.)
|- ( ( ( ph \/ ps ) -> ( ph <-> ps ) ) <-> ( ph <-> ps ) )
Theorem	pm3.44 715	Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
|- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) -> ( ( ps \/ ch ) -> ph ) )
Theorem	jaoi 716	Inference disjoining the antecedents of two implications. (Contributed by NM, 5-Apr-1994.) (Revised by NM, 31-Jan-2015.)
|- ( ph -> ps )   &    |- ( ch -> ps )   =>    |- ( ( ph \/ ch ) -> ps )
Theorem	jaod 717	Deduction disjoining the antecedents of two implications. (Contributed by NM, 18-Aug-1994.) (Revised by NM, 4-Apr-2013.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th -> ch ) )   =>    |- ( ph -> ( ( ps \/ th ) -> ch ) )
Theorem	mpjaod 718	Eliminate a disjunction in a deduction. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th -> ch ) )   &    |- ( ph -> ( ps \/ th ) )   =>    |- ( ph -> ch )
Theorem	jaao 719	Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> ( ta -> ch ) )   =>    |- ( ( ph /\ th ) -> ( ( ps \/ ta ) -> ch ) )
Theorem	jaoa 720	Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> ( ta -> ch ) )   =>    |- ( ( ph \/ th ) -> ( ( ps /\ ta ) -> ch ) )
Theorem	imorr 721	Implication in terms of disjunction. One direction of theorem *4.6 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as seen at imordc 897. (Contributed by Jim Kingdon, 21-Jul-2018.)
|- ( ( -. ph \/ ps ) -> ( ph -> ps ) )
Theorem	pm2.53 722	Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 891). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.)
|- ( ( ph \/ ps ) -> ( -. ph -> ps ) )
Theorem	ori 723	Infer implication from disjunction. (Contributed by NM, 11-Jun-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ph \/ ps )   =>    |- ( -. ph -> ps )
Theorem	ord 724	Deduce implication from disjunction. (Contributed by NM, 18-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ph -> ( ps \/ ch ) )   =>    |- ( ph -> ( -. ps -> ch ) )
Theorem	orel1 725	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.)
|- ( -. ph -> ( ( ph \/ ps ) -> ps ) )
Theorem	orel2 726	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.)
|- ( -. ph -> ( ( ps \/ ph ) -> ps ) )
Theorem	pm1.4 727	Axiom *1.4 of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 15-Nov-2012.)
|- ( ( ph \/ ps ) -> ( ps \/ ph ) )
Theorem	orcom 728	Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Nov-2012.)
|- ( ( ph \/ ps ) <-> ( ps \/ ph ) )
Theorem	orcomd 729	Commutation of disjuncts in consequent. (Contributed by NM, 2-Dec-2010.)
|- ( ph -> ( ps \/ ch ) )   =>    |- ( ph -> ( ch \/ ps ) )
Theorem	orcoms 730	Commutation of disjuncts in antecedent. (Contributed by NM, 2-Dec-2012.)
|- ( ( ph \/ ps ) -> ch )   =>    |- ( ( ps \/ ph ) -> ch )
Theorem	orci 731	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ph   =>    |- ( ph \/ ps )
Theorem	olci 732	Deduction introducing a disjunct. (Contributed by NM, 19-Jan-2008.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ph   =>    |- ( ps \/ ph )
Theorem	orcd 733	Deduction introducing a disjunct. (Contributed by NM, 20-Sep-2007.)
|- ( ph -> ps )   =>    |- ( ph -> ( ps \/ ch ) )
Theorem	olcd 734	Deduction introducing a disjunct. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
|- ( ph -> ps )   =>    |- ( ph -> ( ch \/ ps ) )
Theorem	orcs 735	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 14) -type inference in a proof. (Contributed by NM, 21-Jun-1994.)
|- ( ( ph \/ ps ) -> ch )   =>    |- ( ph -> ch )
Theorem	olcs 736	Deduction eliminating disjunct. (Contributed by NM, 21-Jun-1994.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
|- ( ( ph \/ ps ) -> ch )   =>    |- ( ps -> ch )
Theorem	pm2.07 737	Theorem *2.07 of [WhiteheadRussell] p. 101. (Contributed by NM, 3-Jan-2005.)
|- ( ph -> ( ph \/ ph ) )
Theorem	pm2.45 738	Theorem *2.45 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( -. ( ph \/ ps ) -> -. ph )
Theorem	pm2.46 739	Theorem *2.46 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( -. ( ph \/ ps ) -> -. ps )
Theorem	pm2.47 740	Theorem *2.47 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( -. ( ph \/ ps ) -> ( -. ph \/ ps ) )
Theorem	pm2.48 741	Theorem *2.48 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( -. ( ph \/ ps ) -> ( ph \/ -. ps ) )
Theorem	pm2.49 742	Theorem *2.49 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( -. ( ph \/ ps ) -> ( -. ph \/ -. ps ) )
Theorem	pm2.67 743	Theorem *2.67 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 9-Dec-2012.)
|- ( ( ( ph \/ ps ) -> ps ) -> ( ph -> ps ) )
Theorem	biorf 744	A wff is equivalent to its disjunction with falsehood. Theorem *4.74 of [WhiteheadRussell] p. 121. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
|- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
Theorem	biortn 745	A wff is equivalent to its negated disjunction with falsehood. (Contributed by NM, 9-Jul-2012.)
|- ( ph -> ( ps <-> ( -. ph \/ ps ) ) )
Theorem	biorfi 746	A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.)
|- -. ph   =>    |- ( ps <-> ( ps \/ ph ) )
Theorem	pm2.621 747	Theorem *2.621 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 13-Dec-2013.)
|- ( ( ph -> ps ) -> ( ( ph \/ ps ) -> ps ) )
Theorem	pm2.62 748	Theorem *2.62 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Dec-2013.)
|- ( ( ph \/ ps ) -> ( ( ph -> ps ) -> ps ) )
Theorem	imorri 749	Infer implication from disjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( -. ph \/ ps )   =>    |- ( ph -> ps )
Theorem	pm4.52im 750	One direction of theorem *4.52 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
|- ( ( ph /\ -. ps ) -> -. ( -. ph \/ ps ) )
Theorem	pm4.53r 751	One direction of theorem *4.53 of [WhiteheadRussell] p. 120. The converse also holds in classical logic. (Contributed by Jim Kingdon, 27-Jul-2018.)
|- ( ( -. ph \/ ps ) -> -. ( ph /\ -. ps ) )
Theorem	ioran 752	Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 781, anordc 956, or ianordc 899. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) )
Theorem	pm3.14 753	Theorem *3.14 of [WhiteheadRussell] p. 111. One direction of De Morgan's law). The biconditional holds for decidable propositions as seen at ianordc 899. The converse holds for decidable propositions, as seen at pm3.13dc 959. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )
Theorem	pm3.1 754	Theorem *3.1 of [WhiteheadRussell] p. 111. The converse holds for decidable propositions, as seen at anordc 956. (Contributed by NM, 3-Jan-2005.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ( ph /\ ps ) -> -. ( -. ph \/ -. ps ) )
Theorem	jao 755	Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 4-Apr-2013.)
|- ( ( ph -> ps ) -> ( ( ch -> ps ) -> ( ( ph \/ ch ) -> ps ) ) )
Theorem	pm1.2 756	Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 10-Mar-2013.)
|- ( ( ph \/ ph ) -> ph )
Theorem	oridm 757	Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 16-Apr-2011.) (Proof shortened by Wolf Lammen, 10-Mar-2013.)
|- ( ( ph \/ ph ) <-> ph )
Theorem	pm4.25 758	Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph \/ ph ) )
Theorem	orim12i 759	Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
|- ( ph -> ps )   &    |- ( ch -> th )   =>    |- ( ( ph \/ ch ) -> ( ps \/ th ) )
Theorem	orim1i 760	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
|- ( ph -> ps )   =>    |- ( ( ph \/ ch ) -> ( ps \/ ch ) )
Theorem	orim2i 761	Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
|- ( ph -> ps )   =>    |- ( ( ch \/ ph ) -> ( ch \/ ps ) )
Theorem	orbi2i 762	Inference adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Dec-2012.)
|- ( ph <-> ps )   =>    |- ( ( ch \/ ph ) <-> ( ch \/ ps ) )
Theorem	orbi1i 763	Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- ( ( ph \/ ch ) <-> ( ps \/ ch ) )
Theorem	orbi12i 764	Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph \/ ch ) <-> ( ps \/ th ) )
Theorem	pm1.5 765	Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ps \/ ( ph \/ ch ) ) )
Theorem	or12 766	Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
|- ( ( ph \/ ( ps \/ ch ) ) <-> ( ps \/ ( ph \/ ch ) ) )
Theorem	orass 767	Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ( ps \/ ch ) ) )
Theorem	pm2.31 768	Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ( ph \/ ps ) \/ ch ) )
Theorem	pm2.32 769	Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ph \/ ps ) \/ ch ) -> ( ph \/ ( ps \/ ch ) ) )
Theorem	or32 770	A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
|- ( ( ( ph \/ ps ) \/ ch ) <-> ( ( ph \/ ch ) \/ ps ) )
Theorem	or4 771	Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <-> ( ( ph \/ ch ) \/ ( ps \/ th ) ) )
Theorem	or42 772	Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) <-> ( ( ph \/ ch ) \/ ( th \/ ps ) ) )
Theorem	orordi 773	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
|- ( ( ph \/ ( ps \/ ch ) ) <-> ( ( ph \/ ps ) \/ ( ph \/ ch ) ) )
Theorem	orordir 774	Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
|- ( ( ( ph \/ ps ) \/ ch ) <-> ( ( ph \/ ch ) \/ ( ps \/ ch ) ) )
Theorem	pm2.3 775	Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ps \/ ch ) ) -> ( ph \/ ( ch \/ ps ) ) )
Theorem	pm2.41 776	Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps \/ ( ph \/ ps ) ) -> ( ph \/ ps ) )
Theorem	pm2.42 777	Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( -. ph \/ ( ph -> ps ) ) -> ( ph -> ps ) )
Theorem	pm2.4 778	Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ( ph \/ ps ) ) -> ( ph \/ ps ) )
Theorem	pm4.44 779	Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph \/ ( ph /\ ps ) ) )
Theorem	pm4.56 780	Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
|- ( ( -. ph /\ -. ps ) <-> -. ( ph \/ ps ) )
Theorem	oranim 781	Disjunction in terms of conjunction (DeMorgan's law). One direction of Theorem *4.57 of [WhiteheadRussell] p. 120. The converse does not hold intuitionistically but does hold in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.)
|- ( ( ph \/ ps ) -> -. ( -. ph /\ -. ps ) )
Theorem	pm4.78i 782	Implication distributes over disjunction. One direction of Theorem *4.78 of [WhiteheadRussell] p. 121. The converse holds in classical logic. (Contributed by Jim Kingdon, 15-Jan-2018.)
|- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) -> ( ph -> ( ps \/ ch ) ) )
Theorem	mtord 783	A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ph -> -. ch )   &    |- ( ph -> -. th )   &    |- ( ph -> ( ps -> ( ch \/ th ) ) )   =>    |- ( ph -> -. ps )
Theorem	pm4.45 784	Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
|- ( ph <-> ( ph /\ ( ph \/ ps ) ) )
Theorem	pm3.48 785	Theorem *3.48 of [WhiteheadRussell] p. 114. (Contributed by NM, 28-Jan-1997.) (Revised by NM, 1-Dec-2012.)
|- ( ( ( ph -> ps ) /\ ( ch -> th ) ) -> ( ( ph \/ ch ) -> ( ps \/ th ) ) )
Theorem	orim12d 786	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 10-May-1994.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th -> ta ) )   =>    |- ( ph -> ( ( ps \/ th ) -> ( ch \/ ta ) ) )
Theorem	orim1d 787	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( ps \/ th ) -> ( ch \/ th ) ) )
Theorem	orim2d 788	Disjoin antecedents and consequents in a deduction. (Contributed by NM, 23-Apr-1995.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( th \/ ps ) -> ( th \/ ch ) ) )
Theorem	orim2 789	Axiom *1.6 (Sum) of [WhiteheadRussell] p. 97. (Contributed by NM, 3-Jan-2005.)
|- ( ( ps -> ch ) -> ( ( ph \/ ps ) -> ( ph \/ ch ) ) )
Theorem	orbi2d 790	Deduction adding a left disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th \/ ps ) <-> ( th \/ ch ) ) )
Theorem	orbi1d 791	Deduction adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ th ) ) )
Theorem	orbi1 792	Theorem *4.37 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph <-> ps ) -> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
Theorem	orbi12d 793	Deduction joining two equivalences to form equivalence of disjunctions. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) )
Theorem	pm5.61 794	Theorem *5.61 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 30-Jun-2013.)
|- ( ( ( ph \/ ps ) /\ -. ps ) <-> ( ph /\ -. ps ) )
Theorem	jaoian 795	Inference disjoining the antecedents of two implications. (Contributed by NM, 23-Oct-2005.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( th /\ ps ) -> ch )   =>    |- ( ( ( ph \/ th ) /\ ps ) -> ch )
Theorem	jao1i 796	Add a disjunct in the antecedent of an implication. (Contributed by Rodolfo Medina, 24-Sep-2010.)
|- ( ps -> ( ch -> ph ) )   =>    |- ( ( ph \/ ps ) -> ( ch -> ph ) )
Theorem	jaodan 797	Deduction disjoining the antecedents of two implications. (Contributed by NM, 14-Oct-2005.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ th ) -> ch )   =>    |- ( ( ph /\ ( ps \/ th ) ) -> ch )
Theorem	mpjaodan 798	Eliminate a disjunction in a deduction. A translation of natural deduction rule \/ E ( \/ elimination). (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ th ) -> ch )   &    |- ( ph -> ( ps \/ th ) )   =>    |- ( ph -> ch )
Theorem	pm4.77 799	Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
|- ( ( ( ps -> ph ) /\ ( ch -> ph ) ) <-> ( ( ps \/ ch ) -> ph ) )
Theorem	pm2.63 800	Theorem *2.63 of [WhiteheadRussell] p. 107. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph \/ ps ) -> ( ( -. ph \/ ps ) -> ps ) )