Theorem cbvreuw 3406

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3424 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2371. (Revised by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2137. (Revised by Wolf Lammen, 10-Dec-2024.)

Hypotheses

Ref	Expression
cbvreuw.1	⊢ Ⅎ𝑦𝜑
cbvreuw.2	⊢ Ⅎ𝑥𝜓
cbvreuw.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦

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

Proof of Theorem cbvreuw

Step	Hyp	Ref	Expression
1		cbvreuw.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2		cbvreuw.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		cbvreuw.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvrexw 3304	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
5	1, 2, 3	cbvrmow 3405	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
6	4, 5	anbi12i 627	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 ∧ ∃*𝑦 ∈ 𝐴 𝜓))
7		reu5 3378	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
8		reu5 3378	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 ∧ ∃*𝑦 ∈ 𝐴 𝜓))
9	6, 7, 8	3bitr4i 302	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  Ⅎwnf 1785  ∃wrex 3070  ∃!wreu 3374  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-mo 2534  df-eu 2563  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377

This theorem is referenced by:  cbvrmowOLD  3411  cbvreuvwOLD  3413  reu8nf  3871  poimirlem25  36816