Theorem nfab 2341

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfab.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfab	⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}

Proof of Theorem nfab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfab.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	nfsab 2185	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ 𝜑}
3	2	nfci 2326	1 ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}

Syntax hints:  Ⅎwnf 1471  {cab 2179  Ⅎwnfc 2323

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-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

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-nfc 2325

This theorem is referenced by:  nfaba1  2342  nfrabw  2675  sbcel12g  3096  sbceqg  3097  nfun  3316  nfpw  3615  nfpr  3669  nfop  3821  nfuni  3842  nfint  3881  intab  3900  nfiunxy  3939  nfiinxy  3940  nfiunya  3941  nfiinya  3942  nfiu1  3943  nfii1  3944  nfopab  4098  nfopab1  4099  nfopab2  4100  repizf2  4192  nfdm  4907  fun11iun  5522  eusvobj2  5905  nfoprab1  5968  nfoprab2  5969  nfoprab3  5970  nfoprab  5971  nfrecs  6362  nffrec  6451  nfixpxy  6773  nfixp1  6774  nfwrd  10945