Theorem bnj1454

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj1454.1	⊢ 𝐴 = { 𝑥 ∣ 𝜑 }
Assertion	bnj1454	⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ 𝐴 ↔ [ 𝐵 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1454.1	⊢ 𝐴 = { 𝑥 ∣ 𝜑 }
2	1	eleq2i	⊢ ( 𝐵 ∈ 𝐴 ↔ 𝐵 ∈ { 𝑥 ∣ 𝜑 } )
3		df-sbc	⊢ ( [ 𝐵 / 𝑥 ] 𝜑 ↔ 𝐵 ∈ { 𝑥 ∣ 𝜑 } )
4	3	a1i	⊢ ( 𝐵 ∈ V → ( [ 𝐵 / 𝑥 ] 𝜑 ↔ 𝐵 ∈ { 𝑥 ∣ 𝜑 } ) )
5	2 4	bitr4id	⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ 𝐴 ↔ [ 𝐵 / 𝑥 ] 𝜑 ) )