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Metamath Proof Explorer

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) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 5-Oct-2024)

	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	2	elabg	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )
4	1 3	ax-mp	⊢ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )