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Theorem rmoim 2830
 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 2375 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 imdistan 434 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) ↔ ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
32albii 1411 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
41, 3bitri 183 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
5 moim 2019 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥(𝑥𝐴𝜓) → ∃*𝑥(𝑥𝐴𝜑)))
6 df-rmo 2378 . . 3 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
7 df-rmo 2378 . . 3 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
85, 6, 73imtr4g 204 . 2 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐴𝜓)) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
94, 8sylbi 120 1 (∀𝑥𝐴 (𝜑𝜓) → (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1294   ∈ wcel 1445  ∃*wmo 1956  ∀wral 2370  ∃*wrmo 2373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480 This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-ral 2375  df-rmo 2378 This theorem is referenced by:  rmoimia  2831  disjss2  3847
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