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Theorem reueqd 3419
Description: Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)
Hypothesis
Ref Expression
rmoeqd.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 3415 . 2 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜑))
2 rmoeqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32reubidv 3396 . 2 (𝐴 = 𝐵 → (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥𝐵 𝜓))
41, 3bitrd 279 1 (𝐴 = 𝐵 → (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  ∃!wreu 3376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-mo 2538  df-eu 2567  df-cleq 2727  df-rex 3069  df-rmo 3378  df-reu 3379
This theorem is referenced by:  aceq1  10155
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