Theorem nfeld 2505

Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)

Hypotheses

Ref	Expression
nfeqd.1	⊢ (φ → ℲxA)
nfeqd.2	⊢ (φ → ℲxB)

Assertion

Ref	Expression
nfeld	⊢ (φ → Ⅎx A ∈ B)

Proof of Theorem nfeld

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clel 2349	. 2 ⊢ (A ∈ B ↔ ∃y(y = A ∧ y ∈ B))
2		nfv 1619	. . 3 ⊢ Ⅎyφ
3		nfcvd 2491	. . . . 5 ⊢ (φ → Ⅎxy)
4		nfeqd.1	. . . . 5 ⊢ (φ → ℲxA)
5	3, 4	nfeqd 2504	. . . 4 ⊢ (φ → Ⅎx y = A)
6		nfeqd.2	. . . . 5 ⊢ (φ → ℲxB)
7	6	nfcrd 2503	. . . 4 ⊢ (φ → Ⅎx y ∈ B)
8	5, 7	nfand 1822	. . 3 ⊢ (φ → Ⅎx(y = A ∧ y ∈ B))
9	2, 8	nfexd 1854	. 2 ⊢ (φ → Ⅎx∃y(y = A ∧ y ∈ B))
10	1, 9	nfxfrd 1571	1 ⊢ (φ → Ⅎx A ∈ B)

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

This theorem was proved from 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-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  nfneld  2614  nfrald  2666  ralcom2  2776  nfreud  2784  nfrmod  2785  nfrmo  2787  nfsbc1d  3064  nfsbcd  3067  sbcrext  3120  nfbrd  4683