Theorem cbvrmovw 3402

Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov 3429 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 16-Jun-2017.) (Revised by GG, 30-Sep-2024.)

Hypothesis

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

Assertion

Ref	Expression
cbvrmovw	⊢ (∃*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 cbvrmovw

Step	Hyp	Ref	Expression
1		eleq1w 2823	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvrmovw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 632	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
4	3	cbvmovw 2601	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
5		df-rmo 3379	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-rmo 3379	. 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
7	4, 5, 6	3bitr4i 303	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2107  ∃*wmo 2537  ∃*wrmo 3378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-clel 2815  df-rmo 3379

This theorem is referenced by:  cbvdisjv  5120