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Theorem rmoim 3035
 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 (x A (φψ) → (∃*x A ψ∃*x A φ))

Proof of Theorem rmoim
StepHypRef Expression
1 df-ral 2619 . . 3 (x A (φψ) ↔ x(x A → (φψ)))
2 imdistan 670 . . . 4 ((x A → (φψ)) ↔ ((x A φ) → (x A ψ)))
32albii 1566 . . 3 (x(x A → (φψ)) ↔ x((x A φ) → (x A ψ)))
41, 3bitri 240 . 2 (x A (φψ) ↔ x((x A φ) → (x A ψ)))
5 moim 2250 . . 3 (x((x A φ) → (x A ψ)) → (∃*x(x A ψ) → ∃*x(x A φ)))
6 df-rmo 2622 . . 3 (∃*x A ψ∃*x(x A ψ))
7 df-rmo 2622 . . 3 (∃*x A φ∃*x(x A φ))
85, 6, 73imtr4g 261 . 2 (x((x A φ) → (x A ψ)) → (∃*x A ψ∃*x A φ))
94, 8sylbi 187 1 (x A (φψ) → (∃*x A ψ∃*x A φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2614  ∃*wrmo 2617 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2619  df-rmo 2622 This theorem is referenced by:  rmoimia  3036  2rmorex  3040
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