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Theorem mob2 2864
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 983 . . 3 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → 𝜑)
2 moi2.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
31, 2syl5ibcom 154 . 2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
4 nfs1v 1912 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜑
5 sbequ12 1744 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
64, 5mo4f 2059 . . . . . . 7 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
7 sp 1488 . . . . . . 7 (∀𝑥𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
86, 7sylbi 120 . . . . . 6 (∃*𝑥𝜑 → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
9 nfv 1508 . . . . . . . . . 10 𝑥𝜓
109, 2sbhypf 2735 . . . . . . . . 9 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑𝜓))
1110anbi2d 459 . . . . . . . 8 (𝑦 = 𝐴 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑𝜓)))
12 eqeq2 2149 . . . . . . . 8 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
1311, 12imbi12d 233 . . . . . . 7 (𝑦 = 𝐴 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑𝜓) → 𝑥 = 𝐴)))
1413spcgv 2773 . . . . . 6 (𝐴𝐵 → (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑𝜓) → 𝑥 = 𝐴)))
158, 14syl5 32 . . . . 5 (𝐴𝐵 → (∃*𝑥𝜑 → ((𝜑𝜓) → 𝑥 = 𝐴)))
1615imp 123 . . . 4 ((𝐴𝐵 ∧ ∃*𝑥𝜑) → ((𝜑𝜓) → 𝑥 = 𝐴))
1716expd 256 . . 3 ((𝐴𝐵 ∧ ∃*𝑥𝜑) → (𝜑 → (𝜓𝑥 = 𝐴)))
18173impia 1178 . 2 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝜓𝑥 = 𝐴))
193, 18impbid 128 1 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962  wal 1329   = wceq 1331  wcel 1480  [wsb 1735  ∃*wmo 2000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  moi2  2865  mob  2866
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