Theorem nfel 2381

Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
nfnfc.1	⊢ Ⅎ𝑥𝐴
nfeq.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfel	⊢ Ⅎ𝑥 𝐴 ∈ 𝐵

Proof of Theorem nfel

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clel 2225	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nfcv 2372	. . . . 5 ⊢ Ⅎ𝑥𝑧
3		nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfeq 2380	. . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐴
5		nfeq.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	5	nfcri 2366	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
7	4, 6	nfan 1611	. . 3 ⊢ Ⅎ𝑥(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)
8	7	nfex 1683	. 2 ⊢ Ⅎ𝑥∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)
9	1, 8	nfxfr 1520	1 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵

Syntax hints:   ∧ wa 104   = wceq 1395  Ⅎwnf 1506  ∃wex 1538   ∈ wcel 2200  Ⅎwnfc 2359

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361

This theorem is referenced by:  nfel1  2383  nfel2  2385  nfnel  2502  elabgf  2945  elrabf  2957  sbcel12g  3139  nfdisjv  4070  rabxfrd  4559  ffnfvf  5793  mptelixpg  6879  elabgft1  16100  elabgf2  16102  bj-rspgt  16108