Theorem ralseurals 50603

Description: "All some one" restricted to a class implies "all some" restricted to that class. Restricted counterpart of alseuals 50602. (Contributed by David A. Wheeler, 21-Jul-2026.)

Assertion

Ref	Expression
ralseurals	⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓))

Proof of Theorem ralseurals

Step	Hyp	Ref	Expression
1		reurex 3373	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜑)
2	1	anim2i 628	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑) → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
3		df-ralseu 50600	. 2 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
4		df-rals 50567	. 2 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑥 ∈ 𝐴 𝜑))
5	2, 3, 4	3imtr4i 295	1 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) → ∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 400  ∀wral 3079  ∃wrex 3089  ∃!wreu 3367  ∀∃wrals 50565  ∀∃!wralseu 50598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-eu 2597  df-rex 3090  df-rmo 3369  df-reu 3370  df-rals 50567  df-ralseu 50600

This theorem is referenced by: (None)