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Theorem mob 2839
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 2671 . . . . 5 (𝐵𝐷𝐵 ∈ V)
2 nfcv 2258 . . . . . . . 8 𝑥𝐴
3 nfv 1493 . . . . . . . . . 10 𝑥 𝐵 ∈ V
4 nfmo1 1989 . . . . . . . . . 10 𝑥∃*𝑥𝜑
5 nfv 1493 . . . . . . . . . 10 𝑥𝜓
63, 4, 5nf3an 1530 . . . . . . . . 9 𝑥(𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓)
7 nfv 1493 . . . . . . . . 9 𝑥(𝐴 = 𝐵𝜒)
86, 7nfim 1536 . . . . . . . 8 𝑥((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
9 moi.1 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝜑𝜓))
1093anbi3d 1281 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) ↔ (𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓)))
11 eqeq1 2124 . . . . . . . . . 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 2837 . . . . . . . 8 ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐵𝜒))
162, 8, 13, 15vtoclgf 2718 . . . . . . 7 (𝐴𝐶 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒)))
1716com12 30 . . . . . 6 ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴𝐶 → (𝐴 = 𝐵𝜒)))
18173expib 1169 . . . . 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 1164 1 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 947   = wceq 1316  wcel 1465  ∃*wmo 1978  Vcvv 2660
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  moi  2840  rmob  2973
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