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Theorem reueqdv 3419
Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 1-Sep-2025.)
Hypothesis
Ref Expression
reueqdv.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
reueqdv (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reueqdv
StepHypRef Expression
1 reueqdv.1 . 2 (𝜑𝐴 = 𝐵)
2 reueq1 3414 . 2 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐵 𝜓))
31, 2syl 17 1 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1535  ∃!wreu 3374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-mo 2536  df-eu 2565  df-cleq 2725  df-rex 3067  df-rmo 3376  df-reu 3377
This theorem is referenced by:  upciclem1  48733
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