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Theorem rmoeqd 2640
Description: Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
raleqd.1 (𝐴 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
rmoeqd (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmoeqd
StepHypRef Expression
1 rmoeq1 2632 . 2 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32rmobidv 2622 . 2 (𝐴 = 𝐵 → (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥𝐵 𝜓))
41, 3bitrd 187 1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1332  ∃*wrmo 2420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rmo 2425
This theorem is referenced by: (None)
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