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Mirrors > Home > NFE Home > Th. List > symdifeq2 | GIF version |
Description: Equality law for intersection. (Contributed by SF, 11-Jan-2015.) |
Ref | Expression |
---|---|
symdifeq2 | ⊢ (A = B → (C ⊕ A) = (C ⊕ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq2 3247 | . . . 4 ⊢ (A = B → (C ∖ A) = (C ∖ B)) | |
2 | 1 | compleqd 3245 | . . 3 ⊢ (A = B → ∼ (C ∖ A) = ∼ (C ∖ B)) |
3 | difeq1 3246 | . . . 4 ⊢ (A = B → (A ∖ C) = (B ∖ C)) | |
4 | 3 | compleqd 3245 | . . 3 ⊢ (A = B → ∼ (A ∖ C) = ∼ (B ∖ C)) |
5 | 2, 4 | nineq12d 3242 | . 2 ⊢ (A = B → ( ∼ (C ∖ A) ⩃ ∼ (A ∖ C)) = ( ∼ (C ∖ B) ⩃ ∼ (B ∖ C))) |
6 | df-symdif 3216 | . . 3 ⊢ (C ⊕ A) = ((C ∖ A) ∪ (A ∖ C)) | |
7 | df-un 3214 | . . 3 ⊢ ((C ∖ A) ∪ (A ∖ C)) = ( ∼ (C ∖ A) ⩃ ∼ (A ∖ C)) | |
8 | 6, 7 | eqtri 2373 | . 2 ⊢ (C ⊕ A) = ( ∼ (C ∖ A) ⩃ ∼ (A ∖ C)) |
9 | df-symdif 3216 | . . 3 ⊢ (C ⊕ B) = ((C ∖ B) ∪ (B ∖ C)) | |
10 | df-un 3214 | . . 3 ⊢ ((C ∖ B) ∪ (B ∖ C)) = ( ∼ (C ∖ B) ⩃ ∼ (B ∖ C)) | |
11 | 9, 10 | eqtri 2373 | . 2 ⊢ (C ⊕ B) = ( ∼ (C ∖ B) ⩃ ∼ (B ∖ C)) |
12 | 5, 8, 11 | 3eqtr4g 2410 | 1 ⊢ (A = B → (C ⊕ A) = (C ⊕ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ⩃ cnin 3204 ∼ ccompl 3205 ∖ cdif 3206 ∪ cun 3207 ⊕ csymdif 3209 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 |
This theorem is referenced by: symdifeq12 3250 symdifeq2i 3252 symdifeq2d 3255 |
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