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Mirrors > Home > QLE Home > Th. List > u1lemana | GIF version |
Description: Lemma for Sasaki implication study. (Contributed by NM, 14-Dec-1997.) |
Ref | Expression |
---|---|
u1lemana | ((a →1 b) ∩ a⊥ ) = a⊥ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-i1 44 | . . 3 (a →1 b) = (a⊥ ∪ (a ∩ b)) | |
2 | 1 | ran 78 | . 2 ((a →1 b) ∩ a⊥ ) = ((a⊥ ∪ (a ∩ b)) ∩ a⊥ ) |
3 | ancom 74 | . . 3 ((a⊥ ∪ (a ∩ b)) ∩ a⊥ ) = (a⊥ ∩ (a⊥ ∪ (a ∩ b))) | |
4 | anabs 121 | . . 3 (a⊥ ∩ (a⊥ ∪ (a ∩ b))) = a⊥ | |
5 | 3, 4 | ax-r2 36 | . 2 ((a⊥ ∪ (a ∩ b)) ∩ a⊥ ) = a⊥ |
6 | 2, 5 | ax-r2 36 | 1 ((a →1 b) ∩ a⊥ ) = a⊥ |
Colors of variables: term |
Syntax hints: = wb 1 ⊥ wn 4 ∪ wo 6 ∩ wa 7 →1 wi1 12 |
This theorem was proved from axioms: ax-a1 30 ax-a2 31 ax-a5 34 ax-r1 35 ax-r2 36 ax-r4 37 ax-r5 38 |
This theorem depends on definitions: df-a 40 df-i1 44 |
This theorem is referenced by: u1lemnoa 660 u12lembi 726 u1lem7 772 |
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