Theorem cbvreuw 3392

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3405 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2402. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2178. (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 3306	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)
5	1, 2, 3	cbvrmow 3391	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
6	4, 5	anbi12i 640	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑦 ∈ 𝐴 𝜓 ∧ ∃*𝑦 ∈ 𝐴 𝜓))
7		reu5 3368	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑))
8		reu5 3368	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ (∃𝑦 ∈ 𝐴 𝜓 ∧ ∃*𝑦 ∈ 𝐴 𝜓))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  Ⅎwnf 1816  ∃wrex 3087  ∃!wreu 3364  ∃*wrmo 3365

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  ax-8 2147  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-mo 2565  df-eu 2595  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367

This theorem is used by:  reu8nf  3824  poimirlem25  38543