Theorem elabf 1414

Description: Membership in a class abstraction with implicit substitution.

Hypotheses

Ref	Expression
elabf.1	⊢ (ψ → ∀xψ)
elabf.2	⊢ A ∈ V
elabf.3	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
elabf	⊢ (A ∈ {x∣φ} ↔ ψ)

Distinct variable group(s):   x,A

Proof of Theorem elabf

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbab1 1095	. . . 4 ⊢ (y ∈ {x∣φ} → ∀x y ∈ {x∣φ})
3	1, 2	hbel 1172	. . 3 ⊢ (A ∈ {x∣φ} → ∀x A ∈ {x∣φ})
4		elabf.1	. . 3 ⊢ (ψ → ∀xψ)
5	3, 4	hbbi 705	. 2 ⊢ ((A ∈ {x∣φ} ↔ ψ) → ∀x(A ∈ {x∣φ} ↔ ψ))
6		elabf.2	. 2 ⊢ A ∈ V
7		eleq1 1149	. . . 4 ⊢ (x = A → (x ∈ {x∣φ} ↔ A ∈ {x∣φ}))
8		abid 1094	. . . 4 ⊢ (x ∈ {x∣φ} ↔ φ)
9	7, 8	syl5bbr 412	. . 3 ⊢ (x = A → (φ ↔ A ∈ {x∣φ}))
10		elabf.3	. . 3 ⊢ (x = A → (φ ↔ ψ))
11	9, 10	bitr3d 408	. 2 ⊢ (x = A → (A ∈ {x∣φ} ↔ ψ))
12	5, 6, 11	vtoclef 1392	1 ⊢ (A ∈ {x∣φ} ↔ ψ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  elab 1415  cbvab 1423

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

metamath.org