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Metamath Proof Explorer

Theorem vexwt

Description: A standard theorem of predicate calculus ( stdpc4 ) expressed using class abstractions. Closed form of vexw . (Contributed by BJ, 14-Jun-2019)

		Ref	Expression
	Assertion	vexwt	⊢ ( ∀ 𝑥 𝜑 → 𝑦 ∈ { 𝑥 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		stdpc4	⊢ ( ∀ 𝑥 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 )
2		df-clab	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝑦 / 𝑥 ] 𝜑 )
3	1 2	sylibr	⊢ ( ∀ 𝑥 𝜑 → 𝑦 ∈ { 𝑥 ∣ 𝜑 } )