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Theorem rmob2 3564
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 2949 . . . 4 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
42, 3sylib 208 . . 3 (𝜑 → ∃*𝑥(𝑥𝐴𝜓))
5 rmoi2.4 . . 3 (𝜑𝑥𝐴)
6 rmoi2.5 . . 3 (𝜑𝜓)
7 eleq1 2718 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
8 rmoi2.1 . . . . 5 (𝑥 = 𝐵 → (𝜓𝜒))
97, 8anbi12d 747 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝜓) ↔ (𝐵𝐴𝜒)))
109mob2 3419 . . 3 ((𝐵𝐴 ∧ ∃*𝑥(𝑥𝐴𝜓) ∧ (𝑥𝐴𝜓)) → (𝑥 = 𝐵 ↔ (𝐵𝐴𝜒)))
111, 4, 5, 6, 10syl112anc 1370 . 2 (𝜑 → (𝑥 = 𝐵 ↔ (𝐵𝐴𝜒)))
121, 11mpbirand 529 1 (𝜑 → (𝑥 = 𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  ∃*wmo 2499  ∃*wrmo 2944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rmo 2949  df-v 3233
This theorem is referenced by:  rmoi2  3565
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