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Theorem reueqd 3331
 Description: Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
Hypothesis
Ref Expression
raleqd.1 (𝐴 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
reueqd (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reueqd
StepHypRef Expression
1 reueq1 3325 . 2 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32reubidv 3307 . 2 (𝐴 = 𝐵 → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥𝐵 𝜓))
41, 3bitrd 282 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ∃!wreu 3072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2557  df-eu 2588  df-cleq 2750  df-clel 2830  df-reu 3077 This theorem is referenced by:  aceq1  9590
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