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Mirrors > Home > NFE Home > Th. List > indif2 | GIF version |
Description: Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.) |
Ref | Expression |
---|---|
indif2 | ⊢ (A ∩ (B ∖ C)) = ((A ∩ B) ∖ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 3465 | . 2 ⊢ ((A ∩ B) ∩ (V ∖ C)) = (A ∩ (B ∩ (V ∖ C))) | |
2 | invdif 3496 | . 2 ⊢ ((A ∩ B) ∩ (V ∖ C)) = ((A ∩ B) ∖ C) | |
3 | invdif 3496 | . . 3 ⊢ (B ∩ (V ∖ C)) = (B ∖ C) | |
4 | 3 | ineq2i 3454 | . 2 ⊢ (A ∩ (B ∩ (V ∖ C))) = (A ∩ (B ∖ C)) |
5 | 1, 2, 4 | 3eqtr3ri 2382 | 1 ⊢ (A ∩ (B ∖ C)) = ((A ∩ B) ∖ C) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 Vcvv 2859 ∖ cdif 3206 ∩ cin 3208 |
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-dif 3215 |
This theorem is referenced by: indif1 3499 indifcom 3500 |
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