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Theorem mob 3683
 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 3487 . . . . 5 (𝐵𝐷𝐵 ∈ V)
2 nfv 1915 . . . . . . . . . 10 𝑥 𝐵 ∈ V
3 nfmo1 2640 . . . . . . . . . 10 𝑥∃*𝑥𝜑
4 nfv 1915 . . . . . . . . . 10 𝑥𝜓
52, 3, 4nf3an 1902 . . . . . . . . 9 𝑥(𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓)
6 nfv 1915 . . . . . . . . 9 𝑥(𝐴 = 𝐵𝜒)
75, 6nfim 1897 . . . . . . . 8 𝑥((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
8 moi.1 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝜑𝜓))
983anbi3d 1439 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) ↔ (𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓)))
10 eqeq1 2826 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
1110bibi1d 347 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑥 = 𝐵𝜒) ↔ (𝐴 = 𝐵𝜒)))
129, 11imbi12d 348 . . . . . . . 8 (𝑥 = 𝐴 → (((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐵𝜒)) ↔ ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))))
13 moi.2 . . . . . . . . 9 (𝑥 = 𝐵 → (𝜑𝜒))
1413mob2 3681 . . . . . . . 8 ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐵𝜒))
157, 12, 14vtoclg1f 3541 . . . . . . 7 (𝐴𝐶 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒)))
1615com12 32 . . . . . 6 ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴𝐶 → (𝐴 = 𝐵𝜒)))
17163expib 1119 . . . . 5 (𝐵 ∈ V → ((∃*𝑥𝜑𝜓) → (𝐴𝐶 → (𝐴 = 𝐵𝜒))))
181, 17syl 17 . . . 4 (𝐵𝐷 → ((∃*𝑥𝜑𝜓) → (𝐴𝐶 → (𝐴 = 𝐵𝜒))))
1918com3r 87 . . 3 (𝐴𝐶 → (𝐵𝐷 → ((∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))))
2019imp 410 . 2 ((𝐴𝐶𝐵𝐷) → ((∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒)))
21203impib 1113 1 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  ∃*wmo 2620  Vcvv 3469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471 This theorem is referenced by:  moi  3684  rmob  3846
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