Theorem dfralseu2 51102

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 51069. (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 3077	. . . 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 1851	. . . 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 3366	. . 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 51101	. 2 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
8		df-alseu 51100	. 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 1567   ∈ wcel 2144  ∃!weu 2593  ∀wral 3076  ∃!wreu 3363  ∀∃!walseu 51098  ∀∃!wralseu 51099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3077  df-reu 3366  df-alseu 51100  df-ralseu 51101

This theorem is used by: (None)