Theorem elab

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of Quine p. 44. (Contributed by NM, 1-Aug-1994)

	Ref	Expression
Hypotheses	elab.1	⊢ 𝐴 ∈ V
	elab.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	elab	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		elab.1	⊢ 𝐴 ∈ V
2		elab.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	3 1 2	elabf	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )