Theorem abeq1i 2282

Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.)

Hypothesis

Ref	Expression
abeqri.1	⊢ {𝑥 ∣ 𝜑} = 𝐴

Assertion

Ref	Expression
abeq1i	⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Proof of Theorem abeq1i

Step	Hyp	Ref	Expression
1		abid 2158	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2		abeqri.1	. . 3 ⊢ {𝑥 ∣ 𝜑} = 𝐴
3	2	eleq2i 2237	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴)
4	1, 3	bitr3i 185	1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 104   = wceq 1348   ∈ wcel 2141  {cab 2156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166

This theorem is referenced by: (None)