Theorem cbvreud 37622

Description: Deduction used to change bound variables in a restricted existential uniqueness quantifier. (Contributed by ML, 27-Mar-2021.)

Hypotheses

Ref	Expression
cbvreud.1	⊢ Ⅎ𝑥𝜑
cbvreud.2	⊢ Ⅎ𝑦𝜑
cbvreud.3	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbvreud.4	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbvreud.5	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbvreud	⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))

Distinct variable group:   𝑥,𝐴,𝑦

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

Proof of Theorem cbvreud

Step	Hyp	Ref	Expression
1		cbvreud.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		cbvreud.2	. . 3 ⊢ Ⅎ𝑦𝜑
3		nfvd 1917	. . . 4 ⊢ (𝜑 → Ⅎ𝑦 𝑥 ∈ 𝐴)
4		cbvreud.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝜓)
5	3, 4	nfand 1899	. . 3 ⊢ (𝜑 → Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜓))
6		nfvd 1917	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
7		cbvreud.4	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
8	6, 7	nfand 1899	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜒))
9		eleq1 2825	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10	9	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
11		cbvreud.5	. . . . . 6 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
12	11	imp 406	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
13	10, 12	anbi12d 633	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐴 ∧ 𝜒)))
14	13	ex 412	. . 3 ⊢ (𝜑 → (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐴 ∧ 𝜒))))
15	1, 2, 5, 8, 14	cbveud 37621	. 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜒)))
16		df-reu 3353	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
17		df-reu 3353	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜒))
18	15, 16, 17	3bitr4g 314	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∃!weu 2569  ∃!wreu 3350

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-mo 2540  df-eu 2570  df-cleq 2729  df-clel 2812  df-reu 3353

This theorem is referenced by:  fvineqsneu  37660