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Theorem rmoimi2 2832
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 2019 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜓)) → (∃*𝑥(𝑥𝐵𝜓) → ∃*𝑥(𝑥𝐴𝜑)))
31, 2ax-mp 7 . 2 (∃*𝑥(𝑥𝐵𝜓) → ∃*𝑥(𝑥𝐴𝜑))
4 df-rmo 2378 . 2 (∃*𝑥𝐵 𝜓 ↔ ∃*𝑥(𝑥𝐵𝜓))
5 df-rmo 2378 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
63, 4, 53imtr4i 200 1 (∃*𝑥𝐵 𝜓 → ∃*𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1294  wcel 1445  ∃*wmo 1956  ∃*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-rmo 2378
This theorem is referenced by: (None)
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