Theorem nfdif 3280

Description: Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
nfdif.1	⊢ Ⅎ𝑥𝐴
nfdif.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfdif	⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Proof of Theorem nfdif

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdif2 3161	. 2 ⊢ (𝐴 ∖ 𝐵) = {𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵}
2		nfdif.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
3	2	nfcri 2330	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
4	3	nfn 1669	. . 3 ⊢ Ⅎ𝑥 ¬ 𝑦 ∈ 𝐵
5		nfdif.1	. . 3 ⊢ Ⅎ𝑥𝐴
6	4, 5	nfrabw 2675	. 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ ¬ 𝑦 ∈ 𝐵}
7	1, 6	nfcxfr 2333	1 ⊢ Ⅎ𝑥(𝐴 ∖ 𝐵)

Syntax hints:  ¬ wn 3   ∈ wcel 2164  Ⅎwnfc 2323  {crab 2476   ∖ cdif 3150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-dif 3155

This theorem is referenced by: (None)