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Theorem List for Metamath Proof Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembiimpa 501 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜓) → 𝜒)

Theorembiimpar 502 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜑𝜒) → 𝜓)

Theorembiimpac 503 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜓𝜑) → 𝜒)

Theorembiimparc 504 Importation inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
(𝜑 → (𝜓𝜒))       ((𝜒𝜑) → 𝜓)

Theoremanimorl 505 Conjunction implies disjunction with one common formula (1/4). (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) → (𝜑𝜒))

Theoremanimorr 506 Conjunction implies disjunction with one common formula (2/4). (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) → (𝜒𝜓))

Theoremanimorlr 507 Conjunction implies disjunction with one common formula (3/4). (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) → (𝜒𝜑))

Theoremanimorrl 508 Conjunction implies disjunction with one common formula (4/4). (Contributed by BJ, 4-Oct-2019.)
((𝜑𝜓) → (𝜓𝜒))

Theoremianor 509 Negated conjunction in terms of disjunction (De Morgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120. (Contributed by NM, 14-May-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓))

Theoremanor 510 Conjunction in terms of disjunction (De Morgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Nov-2012.)
((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Theoremioran 511 Negated disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

Theorempm4.52 512 Theorem *4.52 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)
((𝜑 ∧ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓))

Theorempm4.53 513 Theorem *4.53 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜑𝜓))

Theorempm4.54 514 Theorem *4.54 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)
((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))

Theorempm4.55 515 Theorem *4.55 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓))

Theorempm4.56 516 Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))

Theoremoran 517 Disjunction in terms of conjunction (De Morgan's law). Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)
((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))

Theorempm4.57 518 Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ (𝜑𝜓))

Theorempm3.1 519 Theorem *3.1 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜓) → ¬ (¬ 𝜑 ∨ ¬ 𝜓))

Theorempm3.11 520 Theorem *3.11 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
(¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑𝜓))

Theorempm3.12 521 Theorem *3.12 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))

Theorempm3.13 522 Theorem *3.13 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
(¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))

Theorempm3.14 523 Theorem *3.14 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))

Theoremiba 524 Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.)
(𝜑 → (𝜓 ↔ (𝜓𝜑)))

Theoremibar 525 Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.)
(𝜑 → (𝜓 ↔ (𝜑𝜓)))

Theorembiantru 526 A wff is equivalent to its conjunction with truth. (Contributed by NM, 26-May-1993.)
𝜑       (𝜓 ↔ (𝜓𝜑))

Theorembiantrur 527 A wff is equivalent to its conjunction with truth. (Contributed by NM, 3-Aug-1994.)
𝜑       (𝜓 ↔ (𝜑𝜓))

Theorembiantrud 528 A wff is equivalent to its conjunction with truth. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2013.)
(𝜑𝜓)       (𝜑 → (𝜒 ↔ (𝜒𝜓)))

Theorembiantrurd 529 A wff is equivalent to its conjunction with truth. (Contributed by NM, 1-May-1995.) (Proof shortened by Andrew Salmon, 7-May-2011.)
(𝜑𝜓)       (𝜑 → (𝜒 ↔ (𝜓𝜒)))

Theoremmpbirand 530 Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
(𝜑𝜒)    &   (𝜑 → (𝜓 ↔ (𝜒𝜃)))       (𝜑 → (𝜓𝜃))

Theoremjaao 531 Inference conjoining and disjoining the antecedents of two implications. (Contributed by NM, 30-Sep-1999.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Theoremjaoa 532 Inference disjoining and conjoining the antecedents of two implications. (Contributed by Stefan Allan, 1-Nov-2008.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜏𝜒))       ((𝜑𝜃) → ((𝜓𝜏) → 𝜒))

Theorempm3.44 533 Theorem *3.44 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
(((𝜓𝜑) ∧ (𝜒𝜑)) → ((𝜓𝜒) → 𝜑))

Theoremjao 534 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.)
((𝜑𝜓) → ((𝜒𝜓) → ((𝜑𝜒) → 𝜓)))

Theorempm1.2 535 Axiom *1.2 of [WhiteheadRussell] p. 96, which they call "Taut". (Contributed by NM, 3-Jan-2005.)
((𝜑𝜑) → 𝜑)

Theoremoridm 536 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.)
((𝜑𝜑) ↔ 𝜑)

Theorempm4.25 537 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑𝜑))

Theoremorim12i 538 Disjoin antecedents and consequents of two premises. (Contributed by NM, 6-Jun-1994.) (Proof shortened by Wolf Lammen, 25-Jul-2012.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜓𝜃))

Theoremorim1i 539 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))

Theoremorim2i 540 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremorbi2i 541 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.)
(𝜑𝜓)       ((𝜒𝜑) ↔ (𝜒𝜓))

Theoremorbi1i 542 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)       ((𝜑𝜒) ↔ (𝜓𝜒))

Theoremorbi12i 543 Infer the disjunction of two equivalences. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) ↔ (𝜓𝜃))

Theorempm1.5 544 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜓 ∨ (𝜑𝜒)))

Theoremor12 545 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
((𝜑 ∨ (𝜓𝜒)) ↔ (𝜓 ∨ (𝜑𝜒)))

Theoremorass 546 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.)
(((𝜑𝜓) ∨ 𝜒) ↔ (𝜑 ∨ (𝜓𝜒)))

Theorempm2.31 547 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → ((𝜑𝜓) ∨ 𝜒))

Theorempm2.32 548 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
(((𝜑𝜓) ∨ 𝜒) → (𝜑 ∨ (𝜓𝜒)))

Theoremor32 549 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ 𝜓))

Theoremor4 550 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜓𝜃)))

Theoremor42 551 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
(((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∨ (𝜃𝜓)))

Theoremorordi 552 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
((𝜑 ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (𝜑𝜒)))

Theoremorordir 553 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
(((𝜑𝜓) ∨ 𝜒) ↔ ((𝜑𝜒) ∨ (𝜓𝜒)))

Theoremjca 554 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Equivalent to the natural deduction rule I ( introduction), see natded 27230. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
(𝜑𝜓)    &   (𝜑𝜒)       (𝜑 → (𝜓𝜒))

Theoremjcad 555 Deduction conjoining the consequents of two implications. (Contributed by NM, 15-Jul-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremjca31 556 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → ((𝜓𝜒) ∧ 𝜃))

Theoremjca32 557 Join three consequents. (Contributed by FL, 1-Aug-2009.)
(𝜑𝜓)    &   (𝜑𝜒)    &   (𝜑𝜃)       (𝜑 → (𝜓 ∧ (𝜒𝜃)))

Theoremjcai 558 Deduction replacing implication with conjunction. (Contributed by NM, 15-Jul-1993.)
(𝜑𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓𝜒))

Theoremjctil 559 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(𝜑𝜓)    &   𝜒       (𝜑 → (𝜒𝜓))

Theoremjctir 560 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
(𝜑𝜓)    &   𝜒       (𝜑 → (𝜓𝜒))

Theoremjccir 561 Inference conjoining a consequent of a consequent to the right of the consequent in an implication. See also ex-natded5.3i 27236. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by AV, 20-Aug-2019.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑 → (𝜓𝜒))

Theoremjccil 562 Inference conjoining a consequent of a consequent to the left of the consequent in an implication. Remark: One can also prove this theorem using syl 17 and jca 554 (as done in jccir 561), which would be 4 bytes shorter, but one step longer than the current proof. (Proof modification is discouraged.) (Contributed by AV, 20-Aug-2019.)
(𝜑𝜓)    &   (𝜓𝜒)       (𝜑 → (𝜒𝜓))

Theoremjctl 563 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
𝜓       (𝜑 → (𝜓𝜑))

Theoremjctr 564 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
𝜓       (𝜑 → (𝜑𝜓))

Theoremjctild 565 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)       (𝜑 → (𝜓 → (𝜃𝜒)))

Theoremjctird 566 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝜃)       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremsyl6an 567 A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   ((𝜓𝜃) → 𝜏)       (𝜑 → (𝜒𝜏))

Theoremancl 568 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜑𝜓)))

Theoremanclb 569 Conjoin antecedent to left of consequent. Theorem *4.7 of [WhiteheadRussell] p. 120. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜑𝜓)))

Theorempm5.42 570 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
((𝜑 → (𝜓𝜒)) ↔ (𝜑 → (𝜓 → (𝜑𝜒))))

Theoremancr 571 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
((𝜑𝜓) → (𝜑 → (𝜓𝜑)))

Theoremancrb 572 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
((𝜑𝜓) ↔ (𝜑 → (𝜓𝜑)))

Theoremancli 573 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
(𝜑𝜓)       (𝜑 → (𝜑𝜓))

Theoremancri 574 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
(𝜑𝜓)       (𝜑 → (𝜓𝜑))

Theoremancld 575 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜓𝜒)))

Theoremancrd 576 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜒𝜓)))

Theoremanc2l 577 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜑𝜒))))

Theoremanc2r 578 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜒𝜑))))

Theoremanc2li 579 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜑𝜒)))

Theoremanc2ri 580 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → (𝜒𝜑)))

Theorempm3.41 581 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜑𝜒) → ((𝜑𝜓) → 𝜒))

Theorempm3.42 582 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
((𝜓𝜒) → ((𝜑𝜓) → 𝜒))

Theorempm3.4 583 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
((𝜑𝜓) → (𝜑𝜓))

Theorempm4.45im 584 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
(𝜑 ↔ (𝜑 ∧ (𝜓𝜑)))

Theoremanim12d 585 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜃𝜏))       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))

Theoremanim12d1 586 Variant of anim12d 585 where the second implication does not depend on the antecedent. (Contributed by Rodolfo Medina, 12-Oct-2010.)
(𝜑 → (𝜓𝜒))    &   (𝜃𝜏)       (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))

Theoremanim1d 587 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜓𝜃) → (𝜒𝜃)))

Theoremanim2d 588 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 14-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → ((𝜃𝜓) → (𝜃𝜒)))

Theoremanim12i 589 Conjoin antecedents and consequents of two premises. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜓𝜃))

Theoremanim12ci 590 Variant of anim12i 589 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜒𝜃)       ((𝜑𝜒) → (𝜃𝜓))

Theoremanim1i 591 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
(𝜑𝜓)       ((𝜑𝜒) → (𝜓𝜒))

Theoremanim2i 592 Introduce conjunct to both sides of an implication. (Contributed by NM, 3-Jan-1993.)
(𝜑𝜓)       ((𝜒𝜑) → (𝜒𝜓))

Theoremanim12ii 593 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)
(𝜑 → (𝜓𝜒))    &   (𝜃 → (𝜓𝜏))       ((𝜑𝜃) → (𝜓 → (𝜒𝜏)))

Theoremprth 594 Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 585. 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.)
(((𝜑𝜓) ∧ (𝜒𝜃)) → ((𝜑𝜒) → (𝜓𝜃)))

Theorempm2.3 595 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜓𝜒)) → (𝜑 ∨ (𝜒𝜓)))

Theorempm2.41 596 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜓 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm2.42 597 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((¬ 𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm2.4 598 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
((𝜑 ∨ (𝜑𝜓)) → (𝜑𝜓))

Theorempm2.65da 599 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
((𝜑𝜓) → 𝜒)    &   ((𝜑𝜓) → ¬ 𝜒)       (𝜑 → ¬ 𝜓)

Theorempm4.44 600 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
(𝜑 ↔ (𝜑 ∨ (𝜑𝜓)))

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