Theorem abeq1i 2341

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 2217	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
2		abeqri.1	. . 3 ⊢ {𝑥 ∣ 𝜑} = 𝐴
3	2	eleq2i 2296	. 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ 𝐴)
4	1, 3	bitr3i 186	1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1395   ∈ wcel 2200  {cab 2215

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225

This theorem is referenced by: (None)