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Theorem rmob2 3496
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 2903 . . . 4 (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥(𝑥𝐴𝜓))
42, 3sylib 206 . . 3 (𝜑 → ∃*𝑥(𝑥𝐴𝜓))
5 rmoi2.4 . . 3 (𝜑𝑥𝐴)
6 rmoi2.5 . . 3 (𝜑𝜓)
7 eleq1 2675 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
8 rmoi2.1 . . . . 5 (𝑥 = 𝐵 → (𝜓𝜒))
97, 8anbi12d 742 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝜓) ↔ (𝐵𝐴𝜒)))
109mob2 3352 . . 3 ((𝐵𝐴 ∧ ∃*𝑥(𝑥𝐴𝜓) ∧ (𝑥𝐴𝜓)) → (𝑥 = 𝐵 ↔ (𝐵𝐴𝜒)))
111, 4, 5, 6, 10syl112anc 1321 . 2 (𝜑 → (𝑥 = 𝐵 ↔ (𝐵𝐴𝜒)))
121, 11mpbirand 528 1 (𝜑 → (𝑥 = 𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  ∃*wmo 2458  ∃*wrmo 2898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rmo 2903  df-v 3174
This theorem is referenced by:  rmoi2  3497
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