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Mirrors > Home > QLE Home > Th. List > ka4lem | GIF version |
Description: Lemma for KA4 soundness (AND version) - uses OL only. (Contributed by NM, 25-Oct-1997.) |
Ref | Expression |
---|---|
ka4lem | ((a ∩ b)⊥ ∪ ((a ∩ c) ≡ (b ∩ c))) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-a 40 | . . . 4 (a ∩ b) = (a⊥ ∪ b⊥ )⊥ | |
2 | 1 | con2 67 | . . 3 (a ∩ b)⊥ = (a⊥ ∪ b⊥ ) |
3 | df-a 40 | . . . . 5 (a ∩ c) = (a⊥ ∪ c⊥ )⊥ | |
4 | df-a 40 | . . . . 5 (b ∩ c) = (b⊥ ∪ c⊥ )⊥ | |
5 | 3, 4 | 2bi 99 | . . . 4 ((a ∩ c) ≡ (b ∩ c)) = ((a⊥ ∪ c⊥ )⊥ ≡ (b⊥ ∪ c⊥ )⊥ ) |
6 | conb 122 | . . . . 5 ((a⊥ ∪ c⊥ ) ≡ (b⊥ ∪ c⊥ )) = ((a⊥ ∪ c⊥ )⊥ ≡ (b⊥ ∪ c⊥ )⊥ ) | |
7 | 6 | ax-r1 35 | . . . 4 ((a⊥ ∪ c⊥ )⊥ ≡ (b⊥ ∪ c⊥ )⊥ ) = ((a⊥ ∪ c⊥ ) ≡ (b⊥ ∪ c⊥ )) |
8 | 5, 7 | ax-r2 36 | . . 3 ((a ∩ c) ≡ (b ∩ c)) = ((a⊥ ∪ c⊥ ) ≡ (b⊥ ∪ c⊥ )) |
9 | 2, 8 | 2or 72 | . 2 ((a ∩ b)⊥ ∪ ((a ∩ c) ≡ (b ∩ c))) = ((a⊥ ∪ b⊥ ) ∪ ((a⊥ ∪ c⊥ ) ≡ (b⊥ ∪ c⊥ ))) |
10 | ka4lemo 228 | . 2 ((a⊥ ∪ b⊥ ) ∪ ((a⊥ ∪ c⊥ ) ≡ (b⊥ ∪ c⊥ ))) = 1 | |
11 | 9, 10 | ax-r2 36 | 1 ((a ∩ b)⊥ ∪ ((a ∩ c) ≡ (b ∩ c))) = 1 |
Colors of variables: term |
Syntax hints: = wb 1 ⊥ wn 4 ≡ tb 5 ∪ wo 6 ∩ wa 7 1wt 8 |
This theorem was proved from axioms: ax-a1 30 ax-a2 31 ax-a3 32 ax-a4 33 ax-a5 34 ax-r1 35 ax-r2 36 ax-r4 37 ax-r5 38 |
This theorem depends on definitions: df-b 39 df-a 40 df-t 41 df-f 42 df-le1 130 df-le2 131 |
This theorem is referenced by: (None) |
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