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Theorem mob 2861
Description: Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
Hypotheses
Ref Expression
moi.1 (𝑥 = 𝐴 → (𝜑𝜓))
moi.2 (𝑥 = 𝐵 → (𝜑𝜒))
Assertion
Ref Expression
mob (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem mob
StepHypRef Expression
1 elex 2692 . . . . 5 (𝐵𝐷𝐵 ∈ V)
2 nfcv 2279 . . . . . . . 8 𝑥𝐴
3 nfv 1508 . . . . . . . . . 10 𝑥 𝐵 ∈ V
4 nfmo1 2009 . . . . . . . . . 10 𝑥∃*𝑥𝜑
5 nfv 1508 . . . . . . . . . 10 𝑥𝜓
63, 4, 5nf3an 1545 . . . . . . . . 9 𝑥(𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓)
7 nfv 1508 . . . . . . . . 9 𝑥(𝐴 = 𝐵𝜒)
86, 7nfim 1551 . . . . . . . 8 𝑥((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
9 moi.1 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝜑𝜓))
1093anbi3d 1296 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) ↔ (𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓)))
11 eqeq1 2144 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
1211bibi1d 232 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑥 = 𝐵𝜒) ↔ (𝐴 = 𝐵𝜒)))
1310, 12imbi12d 233 . . . . . . . 8 (𝑥 = 𝐴 → (((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐵𝜒)) ↔ ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))))
14 moi.2 . . . . . . . . 9 (𝑥 = 𝐵 → (𝜑𝜒))
1514mob2 2859 . . . . . . . 8 ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐵𝜒))
162, 8, 13, 15vtoclgf 2739 . . . . . . 7 (𝐴𝐶 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒)))
1716com12 30 . . . . . 6 ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴𝐶 → (𝐴 = 𝐵𝜒)))
18173expib 1184 . . . . 5 (𝐵 ∈ V → ((∃*𝑥𝜑𝜓) → (𝐴𝐶 → (𝐴 = 𝐵𝜒))))
191, 18syl 14 . . . 4 (𝐵𝐷 → ((∃*𝑥𝜑𝜓) → (𝐴𝐶 → (𝐴 = 𝐵𝜒))))
2019com3r 79 . . 3 (𝐴𝐶 → (𝐵𝐷 → ((∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))))
2120imp 123 . 2 ((𝐴𝐶𝐵𝐷) → ((∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒)))
22213impib 1179 1 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  ∃*wmo 1998  Vcvv 2681
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 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683
This theorem is referenced by:  moi  2862  rmob  2996
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