Theorem alseubii 50610

Description: Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 50578. (Contributed by David A. Wheeler, 21-Jul-2026.)

Hypotheses

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

Assertion

Ref	Expression
alseubii	⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))

Proof of Theorem alseubii

Step	Hyp	Ref	Expression
1		alseubii.1	. . . . 5 ⊢ (𝜑 ↔ 𝜒)
2		alseubii.2	. . . . 5 ⊢ (𝜓 ↔ 𝜃)
3	1, 2	imbi12i 353	. . . 4 ⊢ ((𝜑 → 𝜓) ↔ (𝜒 → 𝜃))
4	3	albii 1849	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑥(𝜒 → 𝜃))
5	1	eubii 2613	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜒)
6	4, 5	anbi12i 639	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑) ↔ (∀𝑥(𝜒 → 𝜃) ∧ ∃!𝑥𝜒))
7		df-alseu 50599	. 2 ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
8		df-alseu 50599	. 2 ⊢ (∀∃!𝑥(𝜒 → 𝜃) ↔ (∀𝑥(𝜒 → 𝜃) ∧ ∃!𝑥𝜒))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  ∃!weu 2596  ∀∃!walseu 50597

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-alseu 50599

This theorem is referenced by: (None)