Theorem elab3 2993

Description: Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)

Hypotheses

Ref	Expression
elab3.1	⊢ (ψ → A ∈ V)
elab3.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
elab3	⊢ (A ∈ {x ∣ φ} ↔ ψ)

Distinct variable groups:   ψ,x   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem elab3

Step	Hyp	Ref	Expression
1		elab3.1	. 2 ⊢ (ψ → A ∈ V)
2		elab3.2	. . 3 ⊢ (x = A → (φ ↔ ψ))
3	2	elab3g 2992	. 2 ⊢ ((ψ → A ∈ V) → (A ∈ {x ∣ φ} ↔ ψ))
4	1, 3	ax-mp 5	1 ⊢ (A ∈ {x ∣ φ} ↔ ψ)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860

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-12 1925  ax-ext 2334

This theorem 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  fvelrnb  5366  ovelrn  5609