Theorem ralseubii 50611

Description: Congruence for "all some one" restricted to a class. This is the "all some one" counterpart of ralsbii 50579. (Contributed by David A. Wheeler, 21-Jul-2026.)

Hypotheses

Ref	Expression
ralseubii.1	⊢ (𝜑 ↔ 𝜒)
ralseubii.2	⊢ (𝜓 ↔ 𝜃)

Assertion

Ref	Expression
ralseubii	⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥 ∈ 𝐴(𝜒 → 𝜃))

Proof of Theorem ralseubii

Step	Hyp	Ref	Expression
1		ralseubii.1	. . . . 5 ⊢ (𝜑 ↔ 𝜒)
2		ralseubii.2	. . . . 5 ⊢ (𝜓 ↔ 𝜃)
3	1, 2	imbi12i 353	. . . 4 ⊢ ((𝜑 → 𝜓) ↔ (𝜒 → 𝜃))
4	3	ralbii 3111	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥 ∈ 𝐴 (𝜒 → 𝜃))
5	1	reubii 3378	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜒)
6	4, 5	anbi12i 639	. 2 ⊢ ((∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∀𝑥 ∈ 𝐴 (𝜒 → 𝜃) ∧ ∃!𝑥 ∈ 𝐴 𝜒))
7		df-ralseu 50600	. 2 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃!𝑥 ∈ 𝐴 𝜑))
8		df-ralseu 50600	. 2 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜒 → 𝜃) ↔ (∀𝑥 ∈ 𝐴 (𝜒 → 𝜃) ∧ ∃!𝑥 ∈ 𝐴 𝜒))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∀∃!𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃!𝑥 ∈ 𝐴(𝜒 → 𝜃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wral 3079  ∃!wreu 3367  ∀∃!wralseu 50598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-mo 2567  df-eu 2597  df-ral 3080  df-reu 3370  df-ralseu 50600

This theorem is referenced by: (None)