Theorem rmoeqdv 36612

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

Hypothesis

Ref	Expression
rmoeqdv.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem rmoeqdv

Step	Hyp	Ref	Expression
1		rmoeqdv.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		rmoeq1 3407	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓))
3	1, 2	syl 18	1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  ∃*wrmo 3375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-mo 2573  df-cleq 2761  df-rmo 3376

This theorem is referenced by: (None)