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Theorem notrab 3532

Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Assertion**
Ref	Expression
notrab	⊢ (A ∖ {x ∈ A ∣ φ}) = {x ∈ A ∣ ¬ φ}

**Proof of Theorem notrab**
Step	Hyp	Ref	Expression
1		difab 3523	. 2 ⊢ ({x ∣ x ∈ A} ∖ {x ∣ φ}) = {x ∣ (x ∈ A ∧ ¬ φ)}
2		difin 3492	. . 3 ⊢ (A ∖ (A ∩ {x ∣ φ})) = (A ∖ {x ∣ φ})
3		dfrab3 3531	. . . 4 ⊢ {x ∈ A ∣ φ} = (A ∩ {x ∣ φ})
4	3	difeq2i 3382	. . 3 ⊢ (A ∖ {x ∈ A ∣ φ}) = (A ∖ (A ∩ {x ∣ φ}))
5		abid2 2470	. . . 4 ⊢ {x ∣ x ∈ A} = A
6	5	difeq1i 3381	. . 3 ⊢ ({x ∣ x ∈ A} ∖ {x ∣ φ}) = (A ∖ {x ∣ φ})
7	2, 4, 6	3eqtr4i 2383	. 2 ⊢ (A ∖ {x ∈ A ∣ φ}) = ({x ∣ x ∈ A} ∖ {x ∣ φ})
8		df-rab 2623	. 2 ⊢ {x ∈ A ∣ ¬ φ} = {x ∣ (x ∈ A ∧ ¬ φ)}
9	1, 7, 8	3eqtr4i 2383	1 ⊢ (A ∖ {x ∈ A ∣ φ}) = {x ∈ A ∣ ¬ φ}

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 358 = wceq 1642 ∈ wcel 1710 {cab 2339 {crab 2618 ∖ 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-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215

This theorem is referenced by: (None)