Theorem dfralseu2 50888

Description: The bounded "all some one" form is the general form with the class membership folded into the antecedent. This is the "all some one" counterpart of dfrals2 50855. (Contributed by David A. Wheeler, 21-Jul-2026.)

Assertion

Ref	Expression
dfralseu2	⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))

Proof of Theorem dfralseu2

Step	Hyp	Ref	Expression
1		df-ral 3078	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		impexp 456	. . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
3	2	albii 1852	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
4	1, 3	bitr4i 281	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
5		df-reu 3367	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6	4, 5	anbi12i 640	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
7		df-ralseu 50887	. 2 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
8		df-alseu 50886	. 2 ⊢ (∀∃!𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ↔ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  ∃!weu 2594  ∀wral 3077  ∃!wreu 3364  ∀∃!walseu 50884  ∀∃!wralseu 50885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3078  df-reu 3367  df-alseu 50886  df-ralseu 50887

This theorem is used by: (None)