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Theorem rmoan 2739
 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 1969 . . 3 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)))
2 an12 495 . . . 4 ((𝜓 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝜓𝜑)))
32mobii 1937 . . 3 (∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
41, 3sylib 127 . 2 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
5 df-rmo 2314 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
6 df-rmo 2314 . 2 (∃*𝑥𝐴 (𝜓𝜑) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
74, 5, 63imtr4i 190 1 (∃*𝑥𝐴 𝜑 → ∃*𝑥𝐴 (𝜓𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  ∃*wmo 1901  ∃*wrmo 2309 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-rmo 2314 This theorem is referenced by: (None)
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