Theorem rmoeqdv 37131

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 3400	. 2 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓))
3	1, 2	syl 18	1 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569  ∃*wrmo 3368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-mo 2567  df-cleq 2755  df-rmo 3369

This theorem is used by: (None)