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Theorem rmoeqd 3370
 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 3364 . 2 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32rmobidv 3350 . 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  ∃*wrmo 3112 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 2114  ax-9 2122  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-mo 2601  df-cleq 2794  df-clel 2873  df-rmo 3117 This theorem is referenced by: (None)
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