Theorem elrab 2994

Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)

Hypothesis

Ref	Expression
elrab.1	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
elrab	⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ ψ))

Distinct variable groups:   ψ,x   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem elrab

Step	Hyp	Ref	Expression
1		nfcv 2489	. 2 ⊢ ℲxA
2		nfcv 2489	. 2 ⊢ ℲxB
3		nfv 1619	. 2 ⊢ Ⅎxψ
4		elrab.1	. 2 ⊢ (x = A → (φ ↔ ψ))
5	1, 2, 3, 4	elrabf 2993	1 ⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2618

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 2478  df-rab 2623  df-v 2861

This theorem is referenced by:  elrab3  2995  elrab2  2996  ralrab  2998  rexrab  3000  rabsnt  3797  unimax  3925  ssintub  3944  intminss  3952  elpmg  6013  nmembers1lem1  6268  nmembers1lem3  6270