MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mob2 Structured version   Visualization version   GIF version

Theorem mob2 3709
Description: Consequence of "at most one". (Contributed by NM, 2-Jan-2015.)
Hypothesis
Ref Expression
moi2.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
mob2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem mob2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp3 1135 . . 3 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → 𝜑)
2 moi2.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2syl5ibcom 244 . 2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
4 nfv 1910 . . . . . . . . 9 𝑥𝜓
54, 2sbhypf 3530 . . . . . . . 8 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
65anbi2d 628 . . . . . . 7 (𝑦 = 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓)))
7 eqeq2 2738 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
86, 7imbi12d 343 . . . . . 6 (𝑦 = 𝐴 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝐴)))
98spcgv 3582 . . . . 5 (𝐴𝐵 → (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑𝜓) → 𝑥 = 𝐴)))
10 nfs1v 2146 . . . . . . 7 𝑥[𝑦 / 𝑥]𝜑
11 sbequ12 2239 . . . . . . 7 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
1210, 11mo4f 2556 . . . . . 6 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
13 sp 2172 . . . . . 6 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1412, 13sylbi 216 . . . . 5 (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
159, 14impel 504 . . . 4 ((𝐴𝐵 ∧ ∃*𝑥𝜑) → ((𝜑𝜓) → 𝑥 = 𝐴))
1615expd 414 . . 3 ((𝐴𝐵 ∧ ∃*𝑥𝜑) → (𝜑 → (𝜓𝑥 = 𝐴)))
17163impia 1114 . 2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝜓𝑥 = 𝐴))
183, 17impbid 211 1 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084  wal 1532   = wceq 1534  [wsb 2060  wcel 2099  ∃*wmo 2527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464
This theorem is referenced by:  moi2  3710  mob  3711  rmob2  3885
  Copyright terms: Public domain W3C validator