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Theorem rmob2 3824
 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 3117 . . . 4 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
42, 3sylib 221 . . 3 (𝜑 → ∃*𝑥(𝑥𝐴𝜓))
5 rmoi2.4 . . 3 (𝜑𝑥𝐴)
6 rmoi2.5 . . 3 (𝜑𝜓)
7 eleq1 2880 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
8 rmoi2.1 . . . . 5 (𝑥 = 𝐵 → (𝜓𝜒))
97, 8anbi12d 633 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝜓) ↔ (𝐵𝐴𝜒)))
109mob2 3657 . . 3 ((𝐵𝐴 ∧ ∃*𝑥(𝑥𝐴𝜓) ∧ (𝑥𝐴𝜓)) → (𝑥 = 𝐵 ↔ (𝐵𝐴𝜒)))
111, 4, 5, 6, 10syl112anc 1371 . 2 (𝜑 → (𝑥 = 𝐵 ↔ (𝐵𝐴𝜒)))
121, 11mpbirand 706 1 (𝜑 → (𝑥 = 𝐵𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∃*wmo 2599  ∃*wrmo 3112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-clab 2780  df-cleq 2794  df-clel 2873  df-rmo 3117  df-v 3446 This theorem is referenced by:  rmoi2  3825
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