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  3095  sbceqg  3096  nfun  3315  nfpw  3614  nfpr  3668  nfop  3820  nfuni  3841  nfint  3880  intab  3899  nfiunxy  3938  nfiinxy  3939  nfiunya  3940  nfiinya  3941  nfiu1  3942  nfii1  3943  nfopab  4097  nfopab1  4098  nfopab2  4099  repizf2  4191  nfdm  4906  fun11iun  5521  eusvobj2  5904  nfoprab1  5967  nfoprab2  5968  nfoprab3  5969  nfoprab  5970  nfrecs  6360  nffrec  6449  nfixpxy  6771  nfixp1  6772  nfwrd  10942