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Theorem rmoanid 3068
Description: Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.)
Assertion
Ref Expression
rmoanid (∃*𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃*𝑥𝐴 𝜑)

Proof of Theorem rmoanid
StepHypRef Expression
1 anabs5 653 . . 3 ((𝑥𝐴 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴𝜑))
21mobii 2568 . 2 (∃*𝑥(𝑥𝐴 ∧ (𝑥𝐴𝜑)) ↔ ∃*𝑥(𝑥𝐴𝜑))
3 df-rmo 3063 . 2 (∃*𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝑥𝐴𝜑)))
4 df-rmo 3063 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
52, 3, 43bitr4i 294 1 (∃*𝑥𝐴 (𝑥𝐴𝜑) ↔ ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wcel 2155  ∃*wmo 2563  ∃*wrmo 3058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-mo 2565  df-rmo 3063
This theorem is referenced by: (None)
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