Theorem cbvreuvw 2702

Description: Version of cbvreuv 2698 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Gino Giotto, 25-Aug-2024.)

Hypothesis

Ref	Expression
cbvralvw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvreuvw	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvreuvw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2231	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvralvw.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 470	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
4		equequ1 1705	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
5	3, 4	bibi12d 234	. . . . 5 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑧) ↔ ((𝑦 ∈ 𝐴 ∧ 𝜓) ↔ 𝑦 = 𝑧)))
6	5	cbvalvw 1912	. . . 4 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑧) ↔ ∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜓) ↔ 𝑦 = 𝑧))
7	6	exbii 1598	. . 3 ⊢ (∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑧) ↔ ∃𝑧∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜓) ↔ 𝑦 = 𝑧))
8		df-eu 2022	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑧∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝑥 = 𝑧))
9		df-eu 2022	. . 3 ⊢ (∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧∀𝑦((𝑦 ∈ 𝐴 ∧ 𝜓) ↔ 𝑦 = 𝑧))
10	7, 8, 9	3bitr4ri 212	. 2 ⊢ (∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
11		df-reu 2455	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
12		df-reu 2455	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
13	10, 11, 12	3bitr4ri 212	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  ∃wex 1485  ∃!weu 2019   ∈ wcel 2141  ∃!wreu 2450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-eu 2022  df-clel 2166  df-reu 2455

This theorem is referenced by: (None)