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Theorem rmoan 3615
Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmoan (∃*𝑥𝐴 𝜑 → ∃*𝑥𝐴 (𝜓𝜑))

Proof of Theorem rmoan
StepHypRef Expression
1 moan 2699 . . 3 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)))
2 an12 627 . . . 4 ((𝜓 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝜓𝜑)))
32mobii 2637 . . 3 (∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
41, 3sylib 209 . 2 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
5 df-rmo 3115 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
6 df-rmo 3115 . 2 (∃*𝑥𝐴 (𝜓𝜑) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
74, 5, 63imtr4i 283 1 (∃*𝑥𝐴 𝜑 → ∃*𝑥𝐴 (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 2157  ∃*wmo 2633  ∃*wrmo 3110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001
This theorem depends on definitions:  df-bi 198  df-an 385  df-tru 1641  df-ex 1860  df-mo 2635  df-rmo 3115
This theorem is referenced by:  reuxfr2d  5101  reuxfr3d  29678
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