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Theorem mob 3374
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 3201 . . . . 5 (𝐵𝐷𝐵 ∈ V)
2 nfv 1840 . . . . . . . . . 10 𝑥 𝐵 ∈ V
3 nfmo1 2480 . . . . . . . . . 10 𝑥∃*𝑥𝜑
4 nfv 1840 . . . . . . . . . 10 𝑥𝜓
52, 3, 4nf3an 1828 . . . . . . . . 9 𝑥(𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓)
6 nfv 1840 . . . . . . . . 9 𝑥(𝐴 = 𝐵𝜒)
75, 6nfim 1822 . . . . . . . 8 𝑥((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
8 moi.1 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝜑𝜓))
983anbi3d 1402 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) ↔ (𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓)))
10 eqeq1 2625 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
1110bibi1d 333 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝑥 = 𝐵𝜒) ↔ (𝐴 = 𝐵𝜒)))
129, 11imbi12d 334 . . . . . . . 8 (𝑥 = 𝐴 → (((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐵𝜒)) ↔ ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))))
13 moi.2 . . . . . . . . 9 (𝑥 = 𝐵 → (𝜑𝜒))
1413mob2 3372 . . . . . . . 8 ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐵𝜒))
157, 12, 14vtoclg1f 3254 . . . . . . 7 (𝐴𝐶 → ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒)))
1615com12 32 . . . . . 6 ((𝐵 ∈ V ∧ ∃*𝑥𝜑𝜓) → (𝐴𝐶 → (𝐴 = 𝐵𝜒)))
17163expib 1265 . . . . 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 445 . 2 ((𝐴𝐶𝐵𝐷) → ((∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒)))
21203impib 1259 1 (((𝐴𝐶𝐵𝐷) ∧ ∃*𝑥𝜑𝜓) → (𝐴 = 𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  ∃*wmo 2470  Vcvv 3189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191
This theorem is referenced by:  moi  3375  rmob  3514
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