Theorem rabeq2i 2638

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 2166	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑})
3		rabid 2564	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))
4	2, 3	bitri 183	1 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1299   ∈ wcel 1448  {crab 2379

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-rab 2384

This theorem is referenced by:  tfis  4435  fvmptssdm  5437