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Theorem rmoimi2 3738
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 2624 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜓)) → (∃*𝑥(𝑥𝐵𝜓) → ∃*𝑥(𝑥𝐴𝜑)))
31, 2ax-mp 5 . 2 (∃*𝑥(𝑥𝐵𝜓) → ∃*𝑥(𝑥𝐴𝜑))
4 df-rmo 3151 . 2 (∃*𝑥𝐵 𝜓 ↔ ∃*𝑥(𝑥𝐵𝜓))
5 df-rmo 3151 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
63, 4, 53imtr4i 293 1 (∃*𝑥𝐵 𝜓 → ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528  wcel 2107  ∃*wmo 2618  ∃*wrmo 3146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904
This theorem depends on definitions:  df-bi 208  df-ex 1774  df-mo 2620  df-rmo 3151
This theorem is referenced by: (None)
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