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Theorem rmob 2877
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 2331 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 simprl 491 . . . 4 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → 𝐵𝐴)
3 eleq1 2116 . . . 4 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
42, 3syl5ibcom 148 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶𝐶𝐴))
5 simpl 106 . . . 4 ((𝐶𝐴𝜒) → 𝐶𝐴)
65a1i 9 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → ((𝐶𝐴𝜒) → 𝐶𝐴))
7 simplrl 495 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → 𝐵𝐴)
8 simpr 107 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → 𝐶𝐴)
9 simpll 489 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → ∃*𝑥(𝑥𝐴𝜑))
10 simplrr 496 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → 𝜓)
11 eleq1 2116 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
12 rmoi.b . . . . . . 7 (𝑥 = 𝐵 → (𝜑𝜓))
1311, 12anbi12d 450 . . . . . 6 (𝑥 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝐵𝐴𝜓)))
14 eleq1 2116 . . . . . . 7 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
15 rmoi.c . . . . . . 7 (𝑥 = 𝐶 → (𝜑𝜒))
1614, 15anbi12d 450 . . . . . 6 (𝑥 = 𝐶 → ((𝑥𝐴𝜑) ↔ (𝐶𝐴𝜒)))
1713, 16mob 2745 . . . . 5 (((𝐵𝐴𝐶𝐴) ∧ ∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
187, 8, 9, 7, 10, 17syl212anc 1156 . . . 4 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
1918ex 112 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐶𝐴 → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒))))
204, 6, 19pm5.21ndd 631 . 2 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
211, 20sylanb 272 1 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  ∃*wmo 1917  ∃*wrmo 2326
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rmo 2331  df-v 2576
This theorem is referenced by:  rmoi  2878
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