Theorem rmoeqbidv 36573

Description: Formula-building rule for restricted at-most-one quantifier. Deduction form. General version of rmobidv 3382. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
rmoeqbidv.1	⊢ (𝜑 → 𝐴 = 𝐵)
rmoeqbidv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
rmoeqbidv	⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem rmoeqbidv

Step	Hyp	Ref	Expression
1		rmoeqbidv.1	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	eleq2d 2848	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		rmoeqbidv.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	anbi12d 641	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
5	4	mobidv 2576	. 2 ⊢ (𝜑 → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
6		df-rmo 3367	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-rmo 3367	. 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜒 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
8	5, 6, 7	3bitr4g 316	1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∃*wmo 2564  ∃*wrmo 3366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-mo 2566  df-cleq 2754  df-clel 2837  df-rmo 3367

This theorem is referenced by: (None)