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Theorem rmob 3603
 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 2990 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 simprl 811 . . . 4 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → 𝐵𝐴)
3 eleq1 2759 . . . 4 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
42, 3syl5ibcom 235 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶𝐶𝐴))
5 simpl 474 . . . 4 ((𝐶𝐴𝜒) → 𝐶𝐴)
65a1i 11 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → ((𝐶𝐴𝜒) → 𝐶𝐴))
72anim1i 593 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → (𝐵𝐴𝐶𝐴))
8 simpll 807 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → ∃*𝑥(𝑥𝐴𝜑))
9 simplr 809 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → (𝐵𝐴𝜓))
10 eleq1 2759 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
11 rmoi.b . . . . . . 7 (𝑥 = 𝐵 → (𝜑𝜓))
1210, 11anbi12d 749 . . . . . 6 (𝑥 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝐵𝐴𝜓)))
13 eleq1 2759 . . . . . . 7 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
14 rmoi.c . . . . . . 7 (𝑥 = 𝐶 → (𝜑𝜒))
1513, 14anbi12d 749 . . . . . 6 (𝑥 = 𝐶 → ((𝑥𝐴𝜑) ↔ (𝐶𝐴𝜒)))
1612, 15mob 3462 . . . . 5 (((𝐵𝐴𝐶𝐴) ∧ ∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
177, 8, 9, 16syl3anc 1407 . . . 4 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
1817ex 449 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐶𝐴 → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒))))
194, 6, 18pm5.21ndd 368 . 2 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
201, 19sylanb 490 1 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1564   ∈ wcel 2071  ∃*wmo 2540  ∃*wrmo 2985 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-rmo 2990  df-v 3274 This theorem is referenced by:  rmoi  3604
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