Theorem elrab2 2996

Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)

Hypotheses

Ref	Expression
elrab2.1	⊢ (x = A → (φ ↔ ψ))
elrab2.2	⊢ C = {x ∈ B ∣ φ}

Assertion

Ref	Expression
elrab2	⊢ (A ∈ C ↔ (A ∈ B ∧ ψ))

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

Allowed substitution hints:   φ(x)   C(x)

Proof of Theorem elrab2

Step	Hyp	Ref	Expression
1		elrab2.2	. . 3 ⊢ C = {x ∈ B ∣ φ}
2	1	eleq2i 2417	. 2 ⊢ (A ∈ C ↔ A ∈ {x ∈ B ∣ φ})
3		elrab2.1	. . 3 ⊢ (x = A → (φ ↔ ψ))
4	3	elrab 2994	. 2 ⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ ψ))
5	2, 4	bitri 240	1 ⊢ (A ∈ C ↔ (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:  elrabsf  3084  fvmpti  5699  fvmptss2  5725  nenpw1pwlem2  6085