MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmob Structured version   Visualization version   GIF version

Theorem rmob 3731
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 3111 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
2 simprl 778 . . . 4 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → 𝐵𝐴)
3 eleq1 2880 . . . 4 (𝐵 = 𝐶 → (𝐵𝐴𝐶𝐴))
42, 3syl5ibcom 236 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶𝐶𝐴))
5 simpl 470 . . . 4 ((𝐶𝐴𝜒) → 𝐶𝐴)
65a1i 11 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → ((𝐶𝐴𝜒) → 𝐶𝐴))
72anim1i 604 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → (𝐵𝐴𝐶𝐴))
8 simpll 774 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → ∃*𝑥(𝑥𝐴𝜑))
9 simplr 776 . . . . 5 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → (𝐵𝐴𝜓))
10 eleq1 2880 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
11 rmoi.b . . . . . . 7 (𝑥 = 𝐵 → (𝜑𝜓))
1210, 11anbi12d 618 . . . . . 6 (𝑥 = 𝐵 → ((𝑥𝐴𝜑) ↔ (𝐵𝐴𝜓)))
13 eleq1 2880 . . . . . . 7 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
14 rmoi.c . . . . . . 7 (𝑥 = 𝐶 → (𝜑𝜒))
1513, 14anbi12d 618 . . . . . 6 (𝑥 = 𝐶 → ((𝑥𝐴𝜑) ↔ (𝐶𝐴𝜒)))
1612, 15mob 3593 . . . . 5 (((𝐵𝐴𝐶𝐴) ∧ ∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
177, 8, 9, 16syl3anc 1483 . . . 4 (((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) ∧ 𝐶𝐴) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
1817ex 399 . . 3 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐶𝐴 → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒))))
194, 6, 18pm5.21ndd 370 . 2 ((∃*𝑥(𝑥𝐴𝜑) ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
201, 19sylanb 572 1 ((∃*𝑥𝐴 𝜑 ∧ (𝐵𝐴𝜓)) → (𝐵 = 𝐶 ↔ (𝐶𝐴𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  ∃*wmo 2633  ∃*wrmo 3106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rmo 3111  df-v 3400
This theorem is referenced by:  rmoi  3732
  Copyright terms: Public domain W3C validator