Theorem dfrals2

Description: The bounded "all some" form is the general form with the class membership folded into the antecedent. (Contributed by David A. Wheeler, 22-Oct-2018) (Revised by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Assertion	dfrals2	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
2		impexp	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
3	2	albii	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
4	1 3	bitr4i	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )
5		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
6	4 5	anbi12i	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) ↔ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
7		df-rals	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) )
8		df-als	⊢ ( ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ↔ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
9	6 7 8	3bitr4i	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )