Theorem abeq2d 2300

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

Hypothesis

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

Assertion

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

Proof of Theorem abeq2d

Step	Hyp	Ref	Expression
1		abeqd.1	. . 3 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
2	1	eleq2d 2257	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}))
3		abid 2175	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
4	2, 3	bitrdi 196	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1363   ∈ wcel 2158  {cab 2173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183

This theorem is referenced by:  fvelimab  5585  frecsuclem  6421