Theorem bj-reabeq

Description: Relative form of eqabb . (Contributed by BJ, 27-Dec-2023)

	Ref	Expression
Assertion	bj-reabeq	⊢ ( ( 𝑉 ∩ 𝐴 ) = { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝑉 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		dfrab3	⊢ { 𝑥 ∈ 𝑉 ∣ 𝜑 } = ( 𝑉 ∩ { 𝑥 ∣ 𝜑 } )
2	1	eqeq2i	⊢ ( ( 𝑉 ∩ 𝐴 ) = { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ ( 𝑉 ∩ 𝐴 ) = ( 𝑉 ∩ { 𝑥 ∣ 𝜑 } ) )
3		nfcv	⊢ Ⅎ 𝑥 𝐴
4		nfab1	⊢ Ⅎ 𝑥 { 𝑥 ∣ 𝜑 }
5		nfcv	⊢ Ⅎ 𝑥 𝑉
6	3 4 5	bj-rcleqf	⊢ ( ( 𝑉 ∩ 𝐴 ) = ( 𝑉 ∩ { 𝑥 ∣ 𝜑 } ) ↔ ∀ 𝑥 ∈ 𝑉 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ) )
7		abid	⊢ ( 𝑥 ∈ { 𝑥 ∣ 𝜑 } ↔ 𝜑 )
8	7	bibi2i	⊢ ( ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ) ↔ ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )
9	8	ralbii	⊢ ( ∀ 𝑥 ∈ 𝑉 ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ { 𝑥 ∣ 𝜑 } ) ↔ ∀ 𝑥 ∈ 𝑉 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )
10	2 6 9	3bitri	⊢ ( ( 𝑉 ∩ 𝐴 ) = { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ ∀ 𝑥 ∈ 𝑉 ( 𝑥 ∈ 𝐴 ↔ 𝜑 ) )