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Theorem ssrmof 4010
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, 2dfss2f 3935 . . . 4 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
43biimpi 215 . . 3 (𝐴𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
5 pm3.45 623 . . . 4 ((𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
65alimi 1814 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
7 moim 2539 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)) → (∃*𝑥(𝑥𝐵𝜑) → ∃*𝑥(𝑥𝐴𝜑)))
84, 6, 73syl 18 . 2 (𝐴𝐵 → (∃*𝑥(𝑥𝐵𝜑) → ∃*𝑥(𝑥𝐴𝜑)))
9 df-rmo 3352 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
10 df-rmo 3352 . 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 397  wal 1540  wcel 2107  ∃*wmo 2533  wnfc 2884  ∃*wrmo 3351  wss 3911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rmo 3352  df-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by:  2sqreunnlem1  26813  2sqreunnlem2  26819  disjss1f  31536
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