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Mirrors > Home > QLE Home > Th. List > df2c1 | GIF version |
Description: Dual 'commutes' analogue for ≡ analogue of =. (Contributed by NM, 20-Sep-1998.) |
Ref | Expression |
---|---|
df2c1.1 | a = ((a ∪ b) ∩ (a ∪ b⊥ )) |
Ref | Expression |
---|---|
df2c1 | a C b |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df2c1.1 | . . . . 5 a = ((a ∪ b) ∩ (a ∪ b⊥ )) | |
2 | df-a 40 | . . . . . 6 ((a ∪ b) ∩ (a ∪ b⊥ )) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )⊥ | |
3 | anor3 90 | . . . . . . . . 9 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥ | |
4 | anor3 90 | . . . . . . . . 9 (a⊥ ∩ b⊥ ⊥ ) = (a ∪ b⊥ )⊥ | |
5 | 3, 4 | 2or 72 | . . . . . . . 8 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ )) = ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ ) |
6 | 5 | ax-r1 35 | . . . . . . 7 ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ ) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ )) |
7 | 6 | ax-r4 37 | . . . . . 6 ((a ∪ b)⊥ ∪ (a ∪ b⊥ )⊥ )⊥ = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥ |
8 | 2, 7 | ax-r2 36 | . . . . 5 ((a ∪ b) ∩ (a ∪ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥ |
9 | 1, 8 | ax-r2 36 | . . . 4 a = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ ))⊥ |
10 | 9 | con2 67 | . . 3 a⊥ = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ⊥ )) |
11 | 10 | df-c1 132 | . 2 a⊥ C b⊥ |
12 | 11 | comcom5 458 | 1 a C b |
Colors of variables: term |
Syntax hints: = wb 1 C wc 3 ⊥ wn 4 ∪ wo 6 ∩ wa 7 |
This theorem was proved from axioms: ax-a1 30 ax-a2 31 ax-a3 32 ax-a5 34 ax-r1 35 ax-r2 36 ax-r4 37 ax-r5 38 ax-r3 439 |
This theorem depends on definitions: df-b 39 df-a 40 df-t 41 df-f 42 df-le1 130 df-le2 131 df-c1 132 df-c2 133 |
This theorem is referenced by: gsth 489 |
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