Theorem alseubii 17147

Description: Congruence: equivalents may be substituted inside an "all some one". This is the "all some one" counterpart of alsbii 17115. (Contributed by David A. Wheeler, 22-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 239	. . . 4 ⊢ ((𝜑 → 𝜓) ↔ (𝜒 → 𝜃))
4	3	albii 1523	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑥(𝜒 → 𝜃))
5	1	eubii 2095	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥𝜒)
6	4, 5	anbi12i 464	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑) ↔ (∀𝑥(𝜒 → 𝜃) ∧ ∃!𝑥𝜒))
7		df-alseu 17136	. 2 ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃!𝑥𝜑))
8		df-alseu 17136	. 2 ⊢ (∀∃!𝑥(𝜒 → 𝜃) ↔ (∀𝑥(𝜒 → 𝜃) ∧ ∃!𝑥𝜒))
9	6, 7, 8	3bitr4i 212	1 ⊢ (∀∃!𝑥(𝜑 → 𝜓) ↔ ∀∃!𝑥(𝜒 → 𝜃))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400  ∃!weu 2086  ∀∃!walseu 17134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-eu 2089  df-alseu 17136

This theorem is referenced by: (None)