Theorem cleqabri 1177

Description: Equality of a class variable and a class abstraction (inference rule).

Hypothesis

Ref	Expression
cleqabri.1	⊢ {x∣φ} = A

Assertion

Ref	Expression
cleqabri	⊢ (φ ↔ x ∈ A)

Proof of Theorem cleqabri

Step	Hyp	Ref	Expression
1		abid 1094	. 2 ⊢ (x ∈ {x∣φ} ↔ φ)
2		cleqabri.1	. . 3 ⊢ {x∣φ} = A
3	2	eleq2i 1153	. 2 ⊢ (x ∈ {x∣φ} ↔ x ∈ A)
4	1, 3	bitr3 153	1 ⊢ (φ ↔ x ∈ A)

Syntax hints:   ↔ wb 127  {cab 1090   = wceq 1091   ∈ wcel 1092

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

metamath.org