Theorem nfel 2321

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 2166	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵))
2		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑥𝑧
3		nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4	2, 3	nfeq 2320	. . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐴
5		nfeq.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
6	5	nfcri 2306	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵
7	4, 6	nfan 1558	. . 3 ⊢ Ⅎ𝑥(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)
8	7	nfex 1630	. 2 ⊢ Ⅎ𝑥∃𝑧(𝑧 = 𝐴 ∧ 𝑧 ∈ 𝐵)
9	1, 8	nfxfr 1467	1 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵

Syntax hints:   ∧ wa 103   = wceq 1348  Ⅎwnf 1453  ∃wex 1485   ∈ wcel 2141  Ⅎwnfc 2299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301

This theorem is referenced by:  nfel1  2323  nfel2  2325  nfnel  2442  elabgf  2872  elrabf  2884  sbcel12g  3064  nfdisjv  3978  rabxfrd  4454  ffnfvf  5655  mptelixpg  6712  elabgft1  13813  elabgf2  13815  bj-rspgt  13821