Theorem nfel 2348

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 2192	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nfcv 2339	. . . . 5 ⊢ Ⅎ𝑥𝑧
3		nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfeq 2347	. . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐴
5		nfeq.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	5	nfcri 2333	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
7	4, 6	nfan 1579	. . 3 ⊢ Ⅎ𝑥(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)
8	7	nfex 1651	. 2 ⊢ Ⅎ𝑥∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)
9	1, 8	nfxfr 1488	1 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵

Syntax hints:   ∧ wa 104   = wceq 1364  Ⅎwnf 1474  ∃wex 1506   ∈ wcel 2167  Ⅎwnfc 2326

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328

This theorem is referenced by:  nfel1  2350  nfel2  2352  nfnel  2469  elabgf  2906  elrabf  2918  sbcel12g  3099  nfdisjv  4022  rabxfrd  4504  ffnfvf  5721  mptelixpg  6793  elabgft1  15424  elabgf2  15426  bj-rspgt  15432