Theorem nfel 2498

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 2349	. 2 ⊢ (A ∈ B ↔ ∃z(z = A ∧ z ∈ B))
2		nfcv 2490	. . . . 5 ⊢ Ⅎxz
3		nfnfc.1	. . . . 5 ⊢ ℲxA
4	2, 3	nfeq 2497	. . . 4 ⊢ Ⅎx z = A
5		nfeq.2	. . . . 5 ⊢ ℲxB
6	5	nfcri 2484	. . . 4 ⊢ Ⅎx z ∈ B
7	4, 6	nfan 1824	. . 3 ⊢ Ⅎx(z = A ∧ z ∈ B)
8	7	nfex 1843	. 2 ⊢ Ⅎx∃z(z = A ∧ z ∈ B)
9	1, 8	nfxfr 1570	1 ⊢ Ⅎx A ∈ B

This proof depends on syntax axioms:   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is used by:  nfel1  2500  nfel2  2502  nfnel  2612  elabgf  2984  elrabf  2994  sbcel12g  3152  ffnfvf  5429