Theorem nfel 2227

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 2077	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nfcv 2219	. . . . 5 ⊢ Ⅎ𝑥𝑧
3		nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfeq 2226	. . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐴
5		nfeq.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	5	nfcri 2213	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
7	4, 6	nfan 1497	. . 3 ⊢ Ⅎ𝑥(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)
8	7	nfex 1568	. 2 ⊢ Ⅎ𝑥∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)
9	1, 8	nfxfr 1403	1 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵

Syntax hints:   ∧ wa 102   = wceq 1284  Ⅎwnf 1389  ∃wex 1421   ∈ wcel 1433  Ⅎwnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208

This theorem is referenced by:  nfel1  2229  nfel2  2231  nfnel  2346  elabgf  2736  elrabf  2747  sbcel12g  2921  nfdisjv  3778  rabxfrd  4219  ffnfvf  5345  elabgft1  10588  elabgf2  10590  bj-rspgt  10596