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Metamath Proof Explorer

Theorem rabidd

Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of Quine p. 16. (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	rabidd.1	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
	rabidd.2	⊢ ( 𝜑 → 𝜒 )
Assertion	rabidd	⊢ ( 𝜑 → 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		rabidd.1	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
2		rabidd.2	⊢ ( 𝜑 → 𝜒 )
3		rabid	⊢ ( 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜒 } ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) )
4	1 2 3	sylanbrc	⊢ ( 𝜑 → 𝑥 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜒 } )