Theorem rabid2im

Description: One direction of rabid2 is based on fewer axioms. (Contributed by Wolf Lammen, 26-May-2025)

	Ref	Expression
Assertion	rabid2im	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		pm4.71	⊢ ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
2	1	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
3		eqab	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) → 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
4	2 3	sylbi	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
5		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
6		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
7	6	eqeq2i	⊢ ( 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } ↔ 𝐴 = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
8	4 5 7	3imtr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → 𝐴 = { 𝑥 ∈ 𝐴 ∣ 𝜑 } )