Theorem rabexgf

Description: A version of rabexg using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017)

	Ref	Expression
Hypothesis	rabexgf.1	⊢ Ⅎ 𝑥 𝐴
Assertion	rabexgf	⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V )

Proof

Step	Hyp	Ref	Expression
1		rabexgf.1	⊢ Ⅎ 𝑥 𝐴
2		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
3		simpl	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝑥 ∈ 𝐴 )
4	3	ss2abi	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ 𝑥 ∈ 𝐴 }
5	1	abid2f	⊢ { 𝑥 ∣ 𝑥 ∈ 𝐴 } = 𝐴
6	4 5	sseqtri	⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } ⊆ 𝐴
7	2 6	eqsstri	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ 𝐴
8		ssexg	⊢ ( ( { 𝑥 ∈ 𝐴 ∣ 𝜑 } ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉 ) → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V )
9	7 8	mpan	⊢ ( 𝐴 ∈ 𝑉 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } ∈ V )