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Theorem rmob 3005
Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmoi.b (𝑥 = 𝐵 → (𝜑𝜓))
rmoi.c (𝑥 = 𝐶 → (𝜑𝜒))
Assertion
Ref Expression
rmob ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rmob
StepHypRef Expression
1 df-rmo 2425 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 simprl 521 . . . 4 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → 𝐵𝐴)
3 eleq1 2203 . . . 4 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
42, 3syl5ibcom 154 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶𝐶𝐴))
5 simpl 108 . . . 4 ((𝐶𝐴𝜒) → 𝐶𝐴)
65a1i 9 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → ((𝐶𝐴𝜒) → 𝐶𝐴))
7 simplrl 525 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → 𝐵𝐴)
8 simpr 109 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → 𝐶𝐴)
9 simpll 519 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → ∃*𝑥(𝑥𝐴𝜑))
10 simplrr 526 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → 𝜓)
11 eleq1 2203 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
12 rmoi.b . . . . . . 7 (𝑥 = 𝐵 → (𝜑𝜓))
1311, 12anbi12d 465 . . . . . 6 (𝑥 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝐵𝐴𝜓)))
14 eleq1 2203 . . . . . . 7 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
15 rmoi.c . . . . . . 7 (𝑥 = 𝐶 → (𝜑𝜒))
1614, 15anbi12d 465 . . . . . 6 (𝑥 = 𝐶 → ((𝑥𝐴𝜑) ↔ (𝐶𝐴𝜒)))
1713, 16mob 2870 . . . . 5 (((𝐵𝐴𝐶𝐴) ∧ ∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
187, 8, 9, 7, 10, 17syl212anc 1227 . . . 4 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
1918ex 114 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐶𝐴 → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒))))
204, 6, 19pm5.21ndd 695 . 2 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
211, 20sylanb 282 1 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  ∃*wmo 2001  ∃*wrmo 2420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rmo 2425  df-v 2691
This theorem is referenced by:  rmoi  3006
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