Theorem eqabrd 2887

Description: Equality of a class variable and a class abstraction (deduction form of eqabb 2884). (Contributed by NM, 16-Nov-1995.)

Hypothesis

Ref	Expression
eqabrd.1	⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Assertion

Ref	Expression
eqabrd	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Proof of Theorem eqabrd

Step	Hyp	Ref	Expression
1		eqabrd.1	. . 3 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
2	1	eleq2d 2830	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}))
3		abid 2721	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
4	2, 3	bitrdi 287	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  {cab 2717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819

This theorem is referenced by:  eqabri  2888  fvelimab  6994  mapsnend  9101  nosupbnd2  27779  noinfbnd2  27794  fvineqsneu  37377  fvineqsneq  37378  ispridlc  38030  ac6s6  38132  dib1dim  41122  prprspr2  47392