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Theorem rmoeq1 3135
Description: Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
rmoeq1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmoeq1
StepHypRef Expression
1 nfcv 2767 . 2 𝑥𝐴
2 nfcv 2767 . 2 𝑥𝐵
31, 2rmoeq1f 3131 1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  ∃*wrmo 2915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-eu 2478  df-mo 2479  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rmo 2920
This theorem is referenced by:  rmoeqd  3143  poimirlem2  33029
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