This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ssrabeq

Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017)

Ref Expression
Assertion ssrabeq ⊢ V ⊆ x ∈ V | φ ↔ V = x ∈ V | φ

Proof

Step Hyp Ref Expression
1 ssrab2 ⊢ x ∈ V | φ ⊆ V
2 1 biantru ⊢ V ⊆ x ∈ V | φ ↔ V ⊆ x ∈ V | φ ∧ x ∈ V | φ ⊆ V
3 eqss ⊢ V = x ∈ V | φ ↔ V ⊆ x ∈ V | φ ∧ x ∈ V | φ ⊆ V
4 2 3 bitr4i ⊢ V ⊆ x ∈ V | φ ↔ V = x ∈ V | φ