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Mirrors > Home > QLE Home > Th. List > u5lemonb | GIF version |
Description: Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.) |
Ref | Expression |
---|---|
u5lemonb | ((a →5 b) ∪ b⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b⊥ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-i5 48 | . . 3 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) | |
2 | 1 | ax-r5 38 | . 2 ((a →5 b) ∪ b⊥ ) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) |
3 | ax-a3 32 | . . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∩ b⊥ ) ∪ b⊥ )) | |
4 | lear 161 | . . . . 5 (a⊥ ∩ b⊥ ) ≤ b⊥ | |
5 | 4 | df-le2 131 | . . . 4 ((a⊥ ∩ b⊥ ) ∪ b⊥ ) = b⊥ |
6 | 5 | lor 70 | . . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∩ b⊥ ) ∪ b⊥ )) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b⊥ ) |
7 | 3, 6 | ax-r2 36 | . 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b⊥ ) |
8 | 2, 7 | ax-r2 36 | 1 ((a →5 b) ∪ b⊥ ) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b⊥ ) |
Colors of variables: term |
Syntax hints: = wb 1 ⊥ wn 4 ∪ wo 6 ∩ wa 7 →5 wi5 16 |
This theorem was proved from axioms: ax-a2 31 ax-a3 32 ax-a5 34 ax-r1 35 ax-r2 36 ax-r4 37 ax-r5 38 |
This theorem depends on definitions: df-a 40 df-i5 48 df-le1 130 df-le2 131 |
This theorem is referenced by: u5lemnab 654 u5lem3 753 |
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