Theorem rabeq2i 2818

Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)

Hypothesis

Ref	Expression
rabeq2i.1	⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑}

Assertion

Ref	Expression
rabeq2i	⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))

Proof of Theorem rabeq2i

Step	Hyp	Ref	Expression
1		rabeq2i.1	. . 3 ⊢ 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝜑}
2	1	eleq2i 2305	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑})
3		rabid 2727	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
4	2, 3	bitri 184	1 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {crab 2532

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-rab 2537

This theorem is referenced by:  tfis  4725  fvmptssdm  5784  suplocsrlempr  8164  suplocsrlem  8165