Theorem alseubii 50667

Description: Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 50635. (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 1852	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑥(𝜒 → 𝜃))
5	1	eubii 2615	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜒)
6	4, 5	anbi12i 640	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑) ↔ (∀𝑥(𝜒 → 𝜃) ∧ ∃!𝑥𝜒))
7		df-alseu 50656	. 2 ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
8		df-alseu 50656	. 2 ⊢ (∀∃!𝑥(𝜒 → 𝜃) ↔ (∀𝑥(𝜒 → 𝜃) ∧ ∃!𝑥𝜒))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃!weu 2598  ∀∃!walseu 50654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569  df-eu 2599  df-alseu 50656

This theorem is used by: (None)