Theorem List for Quantum Logic Explorer - 601-700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremu2lemaa 601 Lemma for Dishkant implication study. (Contributed by NM, 14-Dec-1997.)
((a2 b) ∩ a) = (ab)

Theoremu3lemaa 602 Lemma for Kalmbach implication study. (Contributed by NM, 14-Dec-1997.)
((a3 b) ∩ a) = (a ∩ (ab))

Theoremu4lemaa 603 Lemma for non-tollens implication study. (Contributed by NM, 14-Dec-1997.)
((a4 b) ∩ a) = (ab)

Theoremu5lemaa 604 Lemma for relevance implication study. (Contributed by NM, 14-Dec-1997.)
((a5 b) ∩ a) = (ab)

Theoremu1lemana 605 Lemma for Sasaki implication study. (Contributed by NM, 14-Dec-1997.)
((a1 b) ∩ a ) = a

Theoremu2lemana 606 Lemma for Dishkant implication study. (Contributed by NM, 14-Dec-1997.)
((a2 b) ∩ a ) = ((ab) ∪ (ab ))

Theoremu3lemana 607 Lemma for Kalmbach implication study. (Contributed by NM, 14-Dec-1997.)
((a3 b) ∩ a ) = ((ab) ∪ (ab ))

Theoremu4lemana 608 Lemma for non-tollens implication study. (Contributed by NM, 14-Dec-1997.)
((a4 b) ∩ a ) = ((ab) ∪ (ab ))

Theoremu5lemana 609 Lemma for relevance implication study. (Contributed by NM, 14-Dec-1997.)
((a5 b) ∩ a ) = ((ab) ∪ (ab ))

Theoremu1lemab 610 Lemma for Sasaki implication study. Equation 4.10 of [MegPav2000] p. 23. This is the second part of the equation. (Contributed by NM, 14-Dec-1997.)
((a1 b) ∩ b) = ((ab) ∪ (ab))

Theoremu2lemab 611 Lemma for Dishkant implication study. (Contributed by NM, 14-Dec-1997.)
((a2 b) ∩ b) = b

Theoremu3lemab 612 Lemma for Kalmbach implication study. (Contributed by NM, 14-Dec-1997.)
((a3 b) ∩ b) = ((ab) ∪ (ab))

Theoremu4lemab 613 Lemma for non-tollens implication study. (Contributed by NM, 14-Dec-1997.)
((a4 b) ∩ b) = ((ab) ∪ (ab))

Theoremu5lemab 614 Lemma for relevance implication study. (Contributed by NM, 14-Dec-1997.)
((a5 b) ∩ b) = ((ab) ∪ (ab))

Theoremu1lemanb 615 Lemma for Sasaki implication study. (Contributed by NM, 14-Dec-1997.)
((a1 b) ∩ b ) = (ab )

Theoremu2lemanb 616 Lemma for Dishkant implication study. (Contributed by NM, 14-Dec-1997.)
((a2 b) ∩ b ) = (ab )

Theoremu3lemanb 617 Lemma for Kalmbach implication study. (Contributed by NM, 14-Dec-1997.)
((a3 b) ∩ b ) = (ab )

Theoremu4lemanb 618 Lemma for non-tollens implication study. (Contributed by NM, 14-Dec-1997.)
((a4 b) ∩ b ) = ((ab) ∩ b )

Theoremu5lemanb 619 Lemma for relevance implication study. (Contributed by NM, 14-Dec-1997.)
((a5 b) ∩ b ) = (ab )

Theoremu1lemoa 620 Lemma for Sasaki implication study. (Contributed by NM, 14-Dec-1997.)
((a1 b) ∪ a) = 1

Theoremu2lemoa 621 Lemma for Dishkant implication study. (Contributed by NM, 14-Dec-1997.)
((a2 b) ∪ a) = 1

Theoremu3lemoa 622 Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)
((a3 b) ∪ a) = (a ∪ ((ab) ∪ (ab )))

Theoremu4lemoa 623 Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)
((a4 b) ∪ a) = 1

Theoremu5lemoa 624 Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.)
((a5 b) ∪ a) = (a ∪ ((ab) ∪ (ab )))

Theoremu1lemona 625 Lemma for Sasaki implication study. (Contributed by NM, 15-Dec-1997.)
((a1 b) ∪ a ) = (a ∪ (ab))

Theoremu2lemona 626 Lemma for Dishkant implication study. (Contributed by NM, 15-Dec-1997.)
((a2 b) ∪ a ) = (ab)

Theoremu3lemona 627 Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)
((a3 b) ∪ a ) = (ab)

Theoremu4lemona 628 Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)
((a4 b) ∪ a ) = (ab)

Theoremu5lemona 629 Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.)
((a5 b) ∪ a ) = (a ∪ (ab))

Theoremu1lemob 630 Lemma for Sasaki implication study. (Contributed by NM, 15-Dec-1997.)
((a1 b) ∪ b) = (ab)

Theoremu2lemob 631 Lemma for Dishkant implication study. (Contributed by NM, 15-Dec-1997.)
((a2 b) ∪ b) = ((ab ) ∪ b)

Theoremu3lemob 632 Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)
((a3 b) ∪ b) = (ab)

Theoremu4lemob 633 Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)
((a4 b) ∪ b) = (ab)

Theoremu5lemob 634 Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.)
((a5 b) ∪ b) = ((ab ) ∪ b)

Theoremu1lemonb 635 Lemma for Sasaki implication study. (Contributed by NM, 15-Dec-1997.)
((a1 b) ∪ b ) = 1

Theoremu2lemonb 636 Lemma for Dishkant implication study. (Contributed by NM, 15-Dec-1997.)
((a2 b) ∪ b ) = 1

Theoremu3lemonb 637 Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)
((a3 b) ∪ b ) = 1

Theoremu4lemonb 638 Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)
((a4 b) ∪ b ) = (((ab) ∪ (ab)) ∪ b )

Theoremu5lemonb 639 Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.)
((a5 b) ∪ b ) = (((ab) ∪ (ab)) ∪ b )

Theoremu1lemnaa 640 Lemma for Sasaki implication study. (Contributed by NM, 15-Dec-1997.)
((a1 b)a) = (a ∩ (ab ))

Theoremu2lemnaa 641 Lemma for Dishkant implication study. (Contributed by NM, 15-Dec-1997.)
((a2 b)a) = (ab )

Theoremu3lemnaa 642 Lemma for Kalmbach implication study. (Contributed by NM, 15-Dec-1997.)
((a3 b)a) = (ab )

Theoremu4lemnaa 643 Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)
((a4 b)a) = (ab )

Theoremu5lemnaa 644 Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.)
((a5 b)a) = (a ∩ (ab ))

Theoremu1lemnana 645 Lemma for Sasaki implication study. (Contributed by NM, 15-Dec-1997.)
((a1 b)a ) = 0

Theoremu2lemnana 646 Lemma for Dishkant implication study. (Contributed by NM, 15-Dec-1997.)
((a2 b)a ) = 0

Theoremu3lemnana 647 Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)
((a3 b)a ) = (a ∩ ((ab) ∩ (ab )))

Theoremu4lemnana 648 Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)
((a4 b)a ) = 0

Theoremu5lemnana 649 Lemma for relevance implication study. (Contributed by NM, 16-Dec-1997.)
((a5 b)a ) = (a ∩ ((ab) ∩ (ab )))

Theoremu1lemnab 650 Lemma for Sasaki implication study. (Contributed by NM, 16-Dec-1997.)
((a1 b)b) = 0

Theoremu2lemnab 651 Lemma for Dishkant implication study. (Contributed by NM, 16-Dec-1997.)
((a2 b)b) = 0

Theoremu3lemnab 652 Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)
((a3 b)b) = 0

Theoremu4lemnab 653 Lemma for non-tollens implication study. (Contributed by NM, 16-Dec-1997.)
((a4 b)b) = (((ab ) ∩ (ab )) ∩ b)

Theoremu5lemnab 654 Lemma for relevance implication study. (Contributed by NM, 16-Dec-1997.)
((a5 b)b) = (((ab ) ∩ (ab )) ∩ b)

Theoremu1lemnanb 655 Lemma for Sasaki implication study. (Contributed by NM, 16-Dec-1997.)
((a1 b)b ) = (ab )

Theoremu2lemnanb 656 Lemma for Dishkant implication study. (Contributed by NM, 16-Dec-1997.)
((a2 b)b ) = ((ab) ∩ b )

Theoremu3lemnanb 657 Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)
((a3 b)b ) = (ab )

Theoremu4lemnanb 658 Lemma for non-tollens implication study. (Contributed by NM, 16-Dec-1997.)
((a4 b)b ) = (ab )

Theoremu5lemnanb 659 Lemma for relevance implication study. (Contributed by NM, 16-Dec-1997.)
((a5 b)b ) = ((ab) ∩ b )

Theoremu1lemnoa 660 Lemma for Sasaki implication study. (Contributed by NM, 16-Dec-1997.)
((a1 b)a) = a

Theoremu2lemnoa 661 Lemma for Dishkant implication study. (Contributed by NM, 16-Dec-1997.)
((a2 b)a) = ((ab) ∩ (ab ))

Theoremu3lemnoa 662 Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)
((a3 b)a) = ((ab) ∩ (ab ))

Theoremu4lemnoa 663 Lemma for non-tollens implication study. (Contributed by NM, 16-Dec-1997.)
((a4 b)a) = ((ab) ∩ (ab ))

Theoremu5lemnoa 664 Lemma for relevance implication study. (Contributed by NM, 16-Dec-1997.)
((a5 b)a) = ((ab) ∩ (ab ))

Theoremu1lemnona 665 Lemma for Sasaki implication study. (Contributed by NM, 16-Dec-1997.)
((a1 b)a ) = (ab )

Theoremu2lemnona 666 Lemma for Dishkant implication study. (Contributed by NM, 16-Dec-1997.)
((a2 b)a ) = (ab )

Theoremu3lemnona 667 Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)
((a3 b)a ) = (a ∪ (ab ))

Theoremu4lemnona 668 Lemma for non-tollens implication study. (Contributed by NM, 16-Dec-1997.)
((a4 b)a ) = (ab )

Theoremu5lemnona 669 Lemma for relevance implication study. (Contributed by NM, 16-Dec-1997.)
((a5 b)a ) = (ab )

Theoremu1lemnob 670 Lemma for Sasaki implication study. (Contributed by NM, 16-Dec-1997.)
((a1 b)b) = (ab)

Theoremu2lemnob 671 Lemma for Dishkant implication study. (Contributed by NM, 16-Dec-1997.)
((a2 b)b) = (ab)

Theoremu3lemnob 672 Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)
((a3 b)b) = (ab)

Theoremu4lemnob 673 Lemma for non-tollens implication study. (Contributed by NM, 16-Dec-1997.)
((a4 b)b) = ((ab ) ∪ b)

Theoremu5lemnob 674 Lemma for relevance implication study. (Contributed by NM, 16-Dec-1997.)
((a5 b)b) = (ab)

Theoremu1lemnonb 675 Lemma for Sasaki implication study. (Contributed by NM, 16-Dec-1997.)
((a1 b)b ) = ((ab ) ∩ (ab ))

Theoremu2lemnonb 676 Lemma for Dishkant implication study. (Contributed by NM, 16-Dec-1997.)
((a2 b)b ) = b

Theoremu3lemnonb 677 Lemma for Kalmbach implication study. (Contributed by NM, 16-Dec-1997.)
((a3 b)b ) = ((ab ) ∩ (ab ))

Theoremu4lemnonb 678 Lemma for non-tollens implication study. (Contributed by NM, 16-Dec-1997.)
((a4 b)b ) = ((ab ) ∩ (ab ))

Theoremu5lemnonb 679 Lemma for relevance implication study. (Contributed by NM, 16-Dec-1997.)
((a5 b)b ) = ((ab ) ∩ (ab ))

Theoremu1lemc1 680 Commutation theorem for Sasaki implication. (Contributed by NM, 14-Dec-1997.)
a C (a1 b)

Theoremu2lemc1 681 Commutation theorem for Dishkant implication. (Contributed by NM, 14-Dec-1997.)
b C (a2 b)

Theoremu3lemc1 682 Commutation theorem for Kalmbach implication. (Contributed by NM, 14-Dec-1997.)
a C (a3 b)

Theoremu4lemc1 683 Commutation theorem for non-tollens implication. (Contributed by NM, 14-Dec-1997.)
b C (a4 b)

Theoremu5lemc1 684 Commutation theorem for relevance implication. (Contributed by NM, 14-Dec-1997.)
a C (a5 b)

Theoremu5lemc1b 685 Commutation theorem for relevance implication. (Contributed by NM, 14-Dec-1997.)
b C (a5 b)

Theoremu1lemc2 686 Commutation theorem for Sasaki implication. (Contributed by NM, 14-Dec-1997.)
a C b    &   a C c       a C (b1 c)

Theoremu2lemc2 687 Commutation theorem for Dishkant implication. (Contributed by NM, 14-Dec-1997.)
a C b    &   a C c       a C (b2 c)

Theoremu3lemc2 688 Commutation theorem for Kalmbach implication. (Contributed by NM, 14-Dec-1997.)
a C b    &   a C c       a C (b3 c)

Theoremu4lemc2 689 Commutation theorem for non-tollens implication. (Contributed by NM, 14-Dec-1997.)
a C b    &   a C c       a C (b4 c)

Theoremu5lemc2 690 Commutation theorem for relevance implication. (Contributed by NM, 14-Dec-1997.)
a C b    &   a C c       a C (b5 c)

Theoremu1lemc3 691 Commutation theorem for Sasaki implication. (Contributed by NM, 14-Dec-1997.)
a C b       a C (b1 a)

Theoremu2lemc3 692 Commutation theorem for Dishkant implication. (Contributed by NM, 14-Dec-1997.)
a C b       a C (b2 a)

Theoremu3lemc3 693 Commutation theorem for Kalmbach implication. (Contributed by NM, 14-Dec-1997.)
a C b       a C (b3 a)

Theoremu4lemc3 694 Commutation theorem for non-tollens implication. (Contributed by NM, 14-Dec-1997.)
a C b       a C (b4 a)

Theoremu5lemc3 695 Commutation theorem for relevance implication. (Contributed by NM, 14-Dec-1997.)
a C b       a C (b5 a)

Theoremu1lemc5 696 Commutation theorem for Sasaki implication. (Contributed by NM, 11-Jan-1998.)
a C b       a C (a1 b)

Theoremu2lemc5 697 Commutation theorem for Dishkant implication. (Contributed by NM, 11-Jan-1998.)
a C b       a C (a2 b)

Theoremu3lemc5 698 Commutation theorem for Kalmbach implication. (Contributed by NM, 11-Jan-1998.)
a C b       a C (a3 b)

Theoremu4lemc5 699 Commutation theorem for non-tollens implication. (Contributed by NM, 11-Jan-1998.)
a C b       a C (a4 b)

Theoremu5lemc5 700 Commutation theorem for relevance implication. (Contributed by NM, 11-Jan-1998.)
a C b       a C (a5 b)

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