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Theorem ssrmof 4051
Description: "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
ssrexf.1 𝑥𝐴
ssrexf.2 𝑥𝐵
Assertion
Ref Expression
ssrmof (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))

Proof of Theorem ssrmof
StepHypRef Expression
1 ssrexf.1 . . . . 5 𝑥𝐴
2 ssrexf.2 . . . . 5 𝑥𝐵
31, 2dfssf 3974 . . . 4 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
43biimpi 216 . . 3 (𝐴𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
5 pm3.45 622 . . . 4 ((𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
65alimi 1811 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
7 moim 2544 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)) → (∃*𝑥(𝑥𝐵𝜑) → ∃*𝑥(𝑥𝐴𝜑)))
84, 6, 73syl 18 . 2 (𝐴𝐵 → (∃*𝑥(𝑥𝐵𝜑) → ∃*𝑥(𝑥𝐴𝜑)))
9 df-rmo 3380 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
10 df-rmo 3380 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
118, 9, 103imtr4g 296 1 (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538  wcel 2108  ∃*wmo 2538  wnfc 2890  ∃*wrmo 3379  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-mo 2540  df-clel 2816  df-nfc 2892  df-rmo 3380  df-ss 3968
This theorem is referenced by:  2sqreunnlem1  27493  2sqreunnlem2  27499  disjss1f  32585  upeu  48928
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