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Theorem mob2 2743
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 917 . . 3 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → 𝜑)
2 moi2.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2syl5ibcom 148 . 2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
4 nfs1v 1831 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜑
5 sbequ12 1670 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
64, 5mo4f 1976 . . . . . . 7 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
7 sp 1417 . . . . . . 7 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
86, 7sylbi 118 . . . . . 6 (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
9 nfv 1437 . . . . . . . . . 10 𝑥𝜓
109, 2sbhypf 2620 . . . . . . . . 9 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
1110anbi2d 445 . . . . . . . 8 (𝑦 = 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓)))
12 eqeq2 2065 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
1311, 12imbi12d 227 . . . . . . 7 (𝑦 = 𝐴 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝐴)))
1413spcgv 2657 . . . . . 6 (𝐴𝐵 → (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑𝜓) → 𝑥 = 𝐴)))
158, 14syl5 32 . . . . 5 (𝐴𝐵 → (∃*𝑥𝜑 → ((𝜑𝜓) → 𝑥 = 𝐴)))
1615imp 119 . . . 4 ((𝐴𝐵 ∧ ∃*𝑥𝜑) → ((𝜑𝜓) → 𝑥 = 𝐴))
1716expd 249 . . 3 ((𝐴𝐵 ∧ ∃*𝑥𝜑) → (𝜑 → (𝜓𝑥 = 𝐴)))
18173impia 1112 . 2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝜓𝑥 = 𝐴))
193, 18impbid 124 1 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896  wal 1257   = wceq 1259  wcel 1409  [wsb 1661  ∃*wmo 1917
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-v 2576
This theorem is referenced by:  moi2  2744  mob  2745
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