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Theorem List for Metamath Proof Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm4.25 501 Theorem *4.25 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  \/  ph ) )
 
Theoremorim12i 502 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 ) )
 
Theoremorim1i 503 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  \/  ch )  ->  ( ps  \/  ch ) )
 
Theoremorim2i 504 Introduce disjunct to both sides of an implication. (Contributed by NM, 6-Jun-1994.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  \/  ph )  ->  ( ch  \/  ps ) )
 
Theoremorbi2i 505 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 ) )
 
Theoremorbi1i 506 Inference adding a right disjunct to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   =>    |-  ( ( ph  \/  ch )  <->  ( ps  \/  ch ) )
 
Theoremorbi12i 507 Infer the disjunction of two equivalences. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  ps )   &    |-  ( ch  <->  th )   =>    |-  ( ( ph  \/  ch )  <->  ( ps  \/  th ) )
 
Theorempm1.5 508 Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ps  \/  ( ph  \/  ch ) ) )
 
Theoremor12 509 Swap two disjuncts. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Nov-2012.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 ) 
 <->  ( ps  \/  ( ph  \/  ch ) ) )
 
Theoremorass 510 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 ) ) )
 
Theorempm2.31 511 Theorem *2.31 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ( ph  \/  ps )  \/ 
 ch ) )
 
Theorempm2.32 512 Theorem *2.32 of [WhiteheadRussell] p. 105. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ch )  ->  ( ph  \/  ( ps  \/  ch )
 ) )
 
Theoremor32 513 A rearrangement of disjuncts. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( ( ph  \/  ps )  \/  ch ) 
 <->  ( ( ph  \/  ch )  \/  ps )
 )
 
Theoremor4 514 Rearrangement of 4 disjuncts. (Contributed by NM, 12-Aug-1994.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th ) )  <-> 
 ( ( ph  \/  ch )  \/  ( ps 
 \/  th ) ) )
 
Theoremor42 515 Rearrangement of 4 disjuncts. (Contributed by NM, 10-Jan-2005.)
 |-  ( ( ( ph  \/  ps )  \/  ( ch  \/  th ) )  <-> 
 ( ( ph  \/  ch )  \/  ( th  \/  ps ) ) )
 
Theoremorordi 516 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 ) 
 <->  ( ( ph  \/  ps )  \/  ( ph  \/  ch ) ) )
 
Theoremorordir 517 Distribution of disjunction over disjunction. (Contributed by NM, 25-Feb-1995.)
 |-  ( ( ( ph  \/  ps )  \/  ch ) 
 <->  ( ( ph  \/  ch )  \/  ( ps 
 \/  ch ) ) )
 
Theoremjca 518 Deduce conjunction of the consequents of two implications ("join consequents with 'and'"). Equivalent to the natural deduction rule  /\ I ( /\ introduction), see natded 20790. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  ( ps  /\  ch ) )
 
Theoremjcad 519 Deduction conjoining the consequents of two implications. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremjca31 520 Join three consequents. (Contributed by Jeff Hankins, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  /\  th ) )
 
Theoremjca32 521 Join three consequents. (Contributed by FL, 1-Aug-2009.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  /\  ( ch  /\  th ) ) )
 
Theoremjcai 522 Deduction replacing implication with conjunction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  /\  ch ) )
 
Theoremjctil 523 Inference conjoining a theorem to left of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
 |-  ( ph  ->  ps )   &    |-  ch   =>    |-  ( ph  ->  ( ch  /\  ps ) )
 
Theoremjctir 524 Inference conjoining a theorem to right of consequent in an implication. (Contributed by NM, 31-Dec-1993.)
 |-  ( ph  ->  ps )   &    |-  ch   =>    |-  ( ph  ->  ( ps  /\  ch ) )
 
Theoremjctl 525 Inference conjoining a theorem to the left of a consequent. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
 |- 
 ps   =>    |-  ( ph  ->  ( ps  /\  ph ) )
 
Theoremjctr 526 Inference conjoining a theorem to the right of a consequent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Oct-2012.)
 |- 
 ps   =>    |-  ( ph  ->  ( ph  /\  ps ) )
 
Theoremjctild 527 Deduction conjoining a theorem to left of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( th  /\  ch ) ) )
 
Theoremjctird 528 Deduction conjoining a theorem to right of consequent in an implication. (Contributed by NM, 21-Apr-2005.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremancl 529 Conjoin antecedent to left of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ( ph  /\  ps ) ) )
 
Theoremanclb 530 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.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  (
 ph  /\  ps )
 ) )
 
Theorempm5.42 531 Theorem *5.42 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 ) 
 <->  ( ph  ->  ( ps  ->  ( ph  /\  ch ) ) ) )
 
Theoremancr 532 Conjoin antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ps )  ->  ( ph  ->  ( ps  /\  ph )
 ) )
 
Theoremancrb 533 Conjoin antecedent to right of consequent. (Contributed by NM, 25-Jul-1999.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
 |-  ( ( ph  ->  ps )  <->  ( ph  ->  ( ps  /\  ph )
 ) )
 
Theoremancli 534 Deduction conjoining antecedent to left of consequent. (Contributed by NM, 12-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ph  /\  ps ) )
 
Theoremancri 535 Deduction conjoining antecedent to right of consequent. (Contributed by NM, 15-Aug-1994.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( ps  /\  ph ) )
 
Theoremancld 536 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ps  /\ 
 ch ) ) )
 
Theoremancrd 537 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\ 
 ps ) ) )
 
Theoremanc2l 538 Conjoin antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 14-Jul-2013.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ( ps  ->  ( ph  /\  ch ) ) ) )
 
Theoremanc2r 539 Conjoin antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.)
 |-  ( ( ph  ->  ( ps  ->  ch )
 )  ->  ( ph  ->  ( ps  ->  ( ch  /\  ph ) ) ) )
 
Theoremanc2li 540 Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 10-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ph  /\ 
 ch ) ) )
 
Theoremanc2ri 541 Deduction conjoining antecedent to right of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  ph ) ) )
 
Theorempm3.41 542 Theorem *3.41 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ch )  ->  ( ( ph  /\  ps )  ->  ch ) )
 
Theorempm3.42 543 Theorem *3.42 of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  ->  ch )  ->  ( ( ph  /\  ps )  ->  ch ) )
 
Theorempm3.4 544 Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113. (Contributed by NM, 31-Jul-1995.)
 |-  ( ( ph  /\  ps )  ->  ( ph  ->  ps ) )
 
Theorempm4.45im 545 Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119. (Contributed by NM, 17-May-1998.)
 |-  ( ph  <->  ( ph  /\  ( ps  ->  ph ) ) )
 
Theoremanim12d 546 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ph  ->  (
 ( ps  /\  th )  ->  ( ch  /\  ta ) ) )
 
Theoremanim1d 547 Add a conjunct to right of antecedent and consequent in a deduction. (Contributed by NM, 3-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( ps  /\  th )  ->  ( ch  /\  th ) ) )
 
Theoremanim2d 548 Add a conjunct to left of antecedent and consequent in a deduction. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ( th  /\  ps )  ->  ( th  /\  ch ) ) )
 
Theoremanim12i 549 Conjoin antecedents and consequents of two premises. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Dec-2013.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   =>    |-  ( ( ph  /\  ch )  ->  ( ps  /\  th ) )
 
Theoremanim12ci 550 Variant of anim12i 549 with commutation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  ps )   &    |-  ( ch  ->  th )   =>    |-  ( ( ph  /\  ch )  ->  ( th  /\  ps ) )
 
Theoremanim1i 551 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ph  /\  ch )  ->  ( ps  /\  ch ) )
 
Theoremanim2i 552 Introduce conjunct to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  (
 ( ch  /\  ph )  ->  ( ch  /\  ps ) )
 
Theoremanim12ii 553 Conjoin antecedents and consequents in a deduction. (Contributed by NM, 11-Nov-2007.) (Proof shortened by Wolf Lammen, 19-Jul-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ps  ->  ta ) )   =>    |-  ( ( ph  /\  th )  ->  ( ps  ->  ( ch  /\  ta )
 ) )
 
Theoremprth 554 Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 546. 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.)
 |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  th ) )  ->  ( ( ph  /\  ch )  ->  ( ps  /\  th ) ) )
 
Theorempm2.3 555 Theorem *2.3 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ps  \/  ch )
 )  ->  ( ph  \/  ( ch  \/  ps ) ) )
 
Theorempm2.41 556 Theorem *2.41 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ps  \/  ( ph  \/  ps )
 )  ->  ( ph  \/  ps ) )
 
Theorempm2.42 557 Theorem *2.42 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( -.  ph  \/  ( ph  ->  ps )
 )  ->  ( ph  ->  ps ) )
 
Theorempm2.4 558 Theorem *2.4 of [WhiteheadRussell] p. 106. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  \/  ( ph  \/  ps )
 )  ->  ( ph  \/  ps ) )
 
Theorempm2.65da 559 Deduction rule for proof by contradiction. (Contributed by NM, 12-Jun-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ps )  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theorempm4.44 560 Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  <->  ( ph  \/  ( ph  /\  ps )
 ) )
 
Theorempm4.14 561 Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  <->  ( ( ph  /\  -.  ch )  ->  -.  ps )
 )
 
Theorempm3.37 562 Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)
 |-  ( ( ( ph  /\ 
 ps )  ->  ch )  ->  ( ( ph  /\  -.  ch )  ->  -.  ps )
 )
 
Theoremnan 563 Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)
 |-  ( ( ph  ->  -.  ( ps  /\  ch ) )  <->  ( ( ph  /\ 
 ps )  ->  -.  ch ) )
 
Theorempm4.15 564 Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)
 |-  ( ( ( ph  /\ 
 ps )  ->  -.  ch ) 
 <->  ( ( ps  /\  ch )  ->  -.  ph )
 )
 
Theorempm4.78 565 Theorem *4.78 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)
 |-  ( ( ( ph  ->  ps )  \/  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  \/  ch )
 ) )
 
Theorempm4.79 566 Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)
 |-  ( ( ( ps 
 ->  ph )  \/  ( ch  ->  ph ) )  <->  ( ( ps 
 /\  ch )  ->  ph )
 )
 
Theorempm4.87 567 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
 |-  ( ( ( ( ( ph  /\  ps )  ->  ch )  <->  ( ph  ->  ( ps  ->  ch )
 ) )  /\  (
 ( ph  ->  ( ps 
 ->  ch ) )  <->  ( ps  ->  (
 ph  ->  ch ) ) ) )  /\  ( ( ps  ->  ( ph  ->  ch ) )  <->  ( ( ps 
 /\  ph )  ->  ch )
 ) )
 
Theorempm3.33 568 Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ch ) )  ->  ( ph  ->  ch )
 )
 
Theorempm3.34 569 Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ps 
 ->  ch )  /\  ( ph  ->  ps ) )  ->  ( ph  ->  ch )
 )
 
Theorempm3.35 570 Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112. (Contributed by NM, 14-Dec-2002.)
 |-  ( ( ph  /\  ( ph  ->  ps ) )  ->  ps )
 
Theorempm5.31 571 Theorem *5.31 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ch  /\  ( ph  ->  ps )
 )  ->  ( ph  ->  ( ps  /\  ch ) ) )
 
Theoremimp4a 572 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch  /\  th )  ->  ta ) ) )
 
Theoremimp4b 573 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  (
 ( ch  /\  th )  ->  ta ) )
 
Theoremimp4c 574 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ( ps  /\  ch )  /\  th )  ->  ta ) )
 
Theoremimp4d 575 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ( ch  /\  th ) ) 
 ->  ta ) )
 
Theoremimp41 576 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theoremimp42 577 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ( ps  /\ 
 ch ) )  /\  th )  ->  ta )
 
Theoremimp43 578 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( (
 ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )
 
Theoremimp44 579 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ( ps  /\  ch )  /\  th )
 )  ->  ta )
 
Theoremimp45 580 An importation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )   =>    |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  ta )
 
Theoremimp5a 581 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ( th  /\  ta )  ->  et ) ) ) )
 
Theoremimp5d 582 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ( ( th  /\ 
 ta )  ->  et )
 )
 
Theoremimp5g 583 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ph  /\  ps )  ->  ( ( ( ch  /\  th )  /\  ta )  ->  et )
 )
 
Theoremimp55 584 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  /\  ta )  ->  et )
 
Theoremimp511 585 An importation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )   =>    |-  ( ( ph  /\  (
 ( ps  /\  ( ch  /\  th ) ) 
 /\  ta ) )  ->  et )
 
Theoremexpimpd 586 Exportation followed by a deduction version of importation. (Contributed by NM, 6-Sep-2008.)
 |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )
 
Theoremexp31 587 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremexp32 588 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
 
Theoremexp4a 589 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ( ( ch 
 /\  th )  ->  ta )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp4b 590 An exportation inference. (Contributed by NM, 26-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
 |-  ( ( ph  /\  ps )  ->  ( ( ch 
 /\  th )  ->  ta )
 )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp4c 591 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  (
 ( ( ps  /\  ch )  /\  th )  ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp4d 592 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ph  ->  (
 ( ps  /\  ( ch  /\  th ) ) 
 ->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp41 593 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp42 594 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  th )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp43 595 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp44 596 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) 
 ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexp45 597 An exportation inference. (Contributed by NM, 26-Apr-1994.)
 |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  ta )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ta )
 ) ) )
 
Theoremexpr 598 Export a wff from a right conjunct. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ps )  ->  ( ch  ->  th ) )
 
Theoremexp5c 599 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  ( ( th  /\ 
 ta )  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
 
Theoremexp53 600 An exportation inference. (Contributed by Jeff Hankins, 30-Aug-2009.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ( ch  /\  th ) )  /\  ta )  ->  et )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( th  ->  ( ta  ->  et ) ) ) ) )
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