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Theorem rmoim 3536
Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
rmoim (∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))

Proof of Theorem rmoim
StepHypRef Expression
1 df-ral 3043 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 imdistan 727 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
32albii 1884 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
41, 3bitri 264 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
5 moim 2645 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥(𝑥𝐴𝜓) → ∃*𝑥(𝑥𝐴𝜑)))
6 df-rmo 3046 . . 3 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
7 df-rmo 3046 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
85, 6, 73imtr4g 285 . 2 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
94, 8sylbi 207 1 (∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1618  wcel 2127  ∃*wmo 2596  wral 3038  ∃*wrmo 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-10 2156  ax-12 2184
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1842  df-nf 1847  df-eu 2599  df-mo 2600  df-ral 3043  df-rmo 3046
This theorem is referenced by:  rmoimia  3537  2rmorex  3541  disjss2  4763  catideu  16508  evlseu  19689  frlmup4  20313  2ndcdisj  21432  poimirlem18  33709  poimirlem21  33712  reuimrmo  41653  2reurex  41656
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