Theorem elabgf

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of Quine p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003) (Revised by Mario Carneiro, 12-Oct-2016)

	Ref	Expression
Hypotheses	elabgf.1	⊢ Ⅎ 𝑥 𝐴
	elabgf.2	⊢ Ⅎ 𝑥 𝜓
	elabgf.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	elabgf	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		elabgf.1	⊢ Ⅎ 𝑥 𝐴
2		elabgf.2	⊢ Ⅎ 𝑥 𝜓
3		elabgf.3	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4		nfab1	⊢ Ⅎ 𝑥 { 𝑥 ∣ 𝜑 }
5	1 4	nfel	⊢ Ⅎ 𝑥 𝐴 ∈ { 𝑥 ∣ 𝜑 }
6	5 2	nfbi	⊢ Ⅎ 𝑥 ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 )
7		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝐴 ∈ { 𝑥 ∣ 𝜑 } ) )
8	7 3	bibi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 ) ↔ ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) ) )
9		abid	⊢ ( 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 )
10	1 6 8 9	vtoclgf	⊢ ( 𝐴 ∈ 𝐵 → ( 𝐴 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜓 ) )