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Theorem rmob2 3873
Description: Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)
Hypotheses
Ref Expression
rmoi2.1 (𝑥 = 𝐵 → (𝜓𝜒))
rmoi2.2 (𝜑𝐵𝐴)
rmoi2.3 (𝜑 → ∃*𝑥𝐴 𝜓)
rmoi2.4 (𝜑𝑥𝐴)
rmoi2.5 (𝜑𝜓)
Assertion
Ref Expression
rmob2 (𝜑 → (𝑥 = 𝐵𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmob2
StepHypRef Expression
1 rmoi2.2 . 2 (𝜑𝐵𝐴)
2 rmoi2.3 . . . 4 (𝜑 → ∃*𝑥𝐴 𝜓)
3 df-rmo 3143 . . . 4 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
42, 3sylib 219 . . 3 (𝜑 → ∃*𝑥(𝑥𝐴𝜓))
5 rmoi2.4 . . 3 (𝜑𝑥𝐴)
6 rmoi2.5 . . 3 (𝜑𝜓)
7 eleq1 2897 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
8 rmoi2.1 . . . . 5 (𝑥 = 𝐵 → (𝜓𝜒))
97, 8anbi12d 630 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝜓) ↔ (𝐵𝐴𝜒)))
109mob2 3703 . . 3 ((𝐵𝐴 ∧ ∃*𝑥(𝑥𝐴𝜓) ∧ (𝑥𝐴𝜓)) → (𝑥 = 𝐵 ↔ (𝐵𝐴𝜒)))
111, 4, 5, 6, 10syl112anc 1366 . 2 (𝜑 → (𝑥 = 𝐵 ↔ (𝐵𝐴𝜒)))
121, 11mpbirand 703 1 (𝜑 → (𝑥 = 𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  ∃*wmo 2613  ∃*wrmo 3138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-clab 2797  df-cleq 2811  df-clel 2890  df-rmo 3143  df-v 3494
This theorem is referenced by:  rmoi2  3874
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