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Theorem rmoim 3734
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 3052 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 imdistan 566 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
32albii 1814 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
41, 3bitri 274 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
5 moim 2533 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥(𝑥𝐴𝜓) → ∃*𝑥(𝑥𝐴𝜑)))
6 df-rmo 3364 . . 3 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
7 df-rmo 3364 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
85, 6, 73imtr4g 295 . 2 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
94, 8sylbi 216 1 (∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1532  wcel 2099  ∃*wmo 2527  wral 3051  ∃*wrmo 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-mo 2529  df-ral 3052  df-rmo 3364
This theorem is referenced by:  rmoimia  3735  reuimrmo  3739  2rmorex  3748  2reurex  3754  disjss2  5121  catideu  17688  rinvmod  19804  frlmup4  21799  evlseu  22098  2ndcdisj  23451  2sqreulem1  27475  2sqreunnlem1  27478  poimirlem18  37339  poimirlem21  37342
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