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Theorem rmoimi2 3542
Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
rmoimi2.1 𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜓))
Assertion
Ref Expression
rmoimi2 (∃*𝑥𝐵 𝜓 → ∃*𝑥𝐴 𝜑)

Proof of Theorem rmoimi2
StepHypRef Expression
1 rmoimi2.1 . . 3 𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜓))
2 moim 2649 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜓)) → (∃*𝑥(𝑥𝐵𝜓) → ∃*𝑥(𝑥𝐴𝜑)))
31, 2ax-mp 5 . 2 (∃*𝑥(𝑥𝐵𝜓) → ∃*𝑥(𝑥𝐴𝜑))
4 df-rmo 3050 . 2 (∃*𝑥𝐵 𝜓 ↔ ∃*𝑥(𝑥𝐵𝜓))
5 df-rmo 3050 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
63, 4, 53imtr4i 281 1 (∃*𝑥𝐵 𝜓 → ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1622  wcel 2131  ∃*wmo 2600  ∃*wrmo 3045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-12 2188
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1846  df-nf 1851  df-eu 2603  df-mo 2604  df-rmo 3050
This theorem is referenced by:  disjin  29698  disjin2  29699
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