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Theorem ssrmo 29917
 Description: "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
ssrmo.1 𝑥𝐴
ssrmo.2 𝑥𝐵
Assertion
Ref Expression
ssrmo (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))

Proof of Theorem ssrmo
StepHypRef Expression
1 ssrmo.1 . . . . 5 𝑥𝐴
2 ssrmo.2 . . . . 5 𝑥𝐵
31, 2dfss2f 3812 . . . 4 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
43biimpi 208 . . 3 (𝐴𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
5 pm3.45 615 . . . 4 ((𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
65alimi 1855 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
7 moim 2556 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)) → (∃*𝑥(𝑥𝐵𝜑) → ∃*𝑥(𝑥𝐴𝜑)))
84, 6, 73syl 18 . 2 (𝐴𝐵 → (∃*𝑥(𝑥𝐵𝜑) → ∃*𝑥(𝑥𝐴𝜑)))
9 df-rmo 3098 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
10 df-rmo 3098 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
118, 9, 103imtr4g 288 1 (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1599   ∈ wcel 2107  ∃*wmo 2549  Ⅎwnfc 2919  ∃*wrmo 3093   ⊆ wss 3792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rmo 3098  df-in 3799  df-ss 3806 This theorem is referenced by:  disjss1f  29966
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