Theorem eliminable-velab 37557

Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment): variable belongs to abstraction. (Contributed by BJ, 30-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
eliminable-velab	⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)

Proof of Theorem eliminable-velab

Step	Hyp	Ref	Expression
1		df-clab 2744	1 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2099   ∈ wcel 2146  {cab 2743

This proof depends on definitions:  df-clab 2744

This theorem is used by:  eliminable-veqab  37558  eliminable-abeqv  37559  eliminable-abeqab  37560  eliminable-abelab  37562