Theorem cbvrmow 3394

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmo 3409 with a disjoint variable condition, which does not require ax-10 2177, ax-13 2405. (Contributed by NM, 16-Jun-2017.) Avoid ax-10 2177, ax-13 2405. (Revised by GG, 23-May-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbvrmow	⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑥,𝑦,𝐴

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

Proof of Theorem cbvrmow

Step	Hyp	Ref	Expression
1		nfv 1936	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
2		cbvrmow.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3	1, 2	nfan 1921	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
4		nfv 1936	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
5		cbvrmow.2	. . . 4 ⊢ Ⅎ𝑥𝜓
6	4, 5	nfan 1921	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)
7		eleq1w 2847	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
8		cbvrmow.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 641	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
10	3, 6, 9	cbvmow 2632	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
11		df-rmo 3369	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
12		df-rmo 3369	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
13	10, 11, 12	3bitr4i 305	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  Ⅎwnf 1805   ∈ wcel 2144  ∃*wmo 2566  ∃*wrmo 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-11 2193  ax-12 2214

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-mo 2568  df-clel 2839  df-rmo 3369

This theorem is referenced by:  cbvreuw  3395  cbvdisj  5079