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Theorem mob2 3686
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 1134 . . 3 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → 𝜑)
2 moi2.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2syl5ibcom 247 . 2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
4 nfv 1915 . . . . . . . . 9 𝑥𝜓
54, 2sbhypf 3531 . . . . . . . 8 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
65anbi2d 630 . . . . . . 7 (𝑦 = 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓)))
7 eqeq2 2832 . . . . . . 7 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
86, 7imbi12d 347 . . . . . 6 (𝑦 = 𝐴 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝐴)))
98spcgv 3574 . . . . 5 (𝐴𝐵 → (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑𝜓) → 𝑥 = 𝐴)))
10 nfs1v 2160 . . . . . . 7 𝑥[𝑦 / 𝑥]𝜑
11 sbequ12 2253 . . . . . . 7 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
1210, 11mo4f 2650 . . . . . 6 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
13 sp 2182 . . . . . 6 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
1412, 13sylbi 219 . . . . 5 (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
159, 14impel 508 . . . 4 ((𝐴𝐵 ∧ ∃*𝑥𝜑) → ((𝜑𝜓) → 𝑥 = 𝐴))
1615expd 418 . . 3 ((𝐴𝐵 ∧ ∃*𝑥𝜑) → (𝜑 → (𝜓𝑥 = 𝐴)))
17163impia 1113 . 2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝜓𝑥 = 𝐴))
183, 17impbid 214 1 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083  wal 1535   = wceq 1537  [wsb 2069  wcel 2114  ∃*wmo 2620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3475
This theorem is referenced by:  moi2  3687  mob  3688  rmob2  3853
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