Theorem abeq2d 2901

Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2899). (Contributed by NM, 16-Nov-1995.)

Hypothesis

Ref	Expression
abeq2d.1	⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})

Assertion

Ref	Expression
abeq2d	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Proof of Theorem abeq2d

Step	Hyp	Ref	Expression
1		abeq2d.1	. . 3 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
2	1	eleq2d 2853	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}))
3		abid 2764	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
4	2, 3	syl6bb 279	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1508   ∈ wcel 2051  {cab 2760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-12 2107  ax-ext 2752

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-sb 2017  df-clab 2761  df-cleq 2773  df-clel 2848

This theorem is referenced by:  abeq2i  2902  fvelimab  6572  mapsnend  8391  nosupbnd2  32777  fvineqsneu  34173  fvineqsneq  34174  ispridlc  34830  ac6s6  34934  dib1dim  37786  prprspr2  43083