Theorem abeq2d 2290

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 2247	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}))
3		abid 2165	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
4	2, 3	bitrdi 196	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173

This theorem is referenced by:  fvelimab  5573  frecsuclem  6407