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Theorem rmoim 3394
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 2917 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 imdistan 724 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
32albii 1744 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
41, 3bitri 264 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
5 moim 2523 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥(𝑥𝐴𝜓) → ∃*𝑥(𝑥𝐴𝜑)))
6 df-rmo 2920 . . 3 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
7 df-rmo 2920 . . 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 384  wal 1478  wcel 1992  ∃*wmo 2475  wral 2912  ∃*wrmo 2915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-12 2049
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-eu 2478  df-mo 2479  df-ral 2917  df-rmo 2920
This theorem is referenced by:  rmoimia  3395  2rmorex  3399  disjss2  4591  catideu  16252  evlseu  19430  frlmup4  20054  2ndcdisj  21164  poimirlem18  33045  poimirlem21  33048  reuimrmo  40469  2reurex  40472
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