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Theorem rmoeqd 3126
 Description: Equality deduction for restricted uniqueness 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 3118 . 2 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32rmobidv 3108 . 2 (𝐴 = 𝐵 → (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥𝐵 𝜓))
41, 3bitrd 267 1 (𝐴 = 𝐵 → (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥𝐵 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃*wrmo 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904 This theorem is referenced by: (None)
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