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Theorem moi2 2920
Description: Consequence of "at most one". (Contributed by NM, 29-Jun-2008.)
Hypothesis
Ref Expression
moi2.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
moi2 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem moi2
StepHypRef Expression
1 moi2.1 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
21mob2 2919 . . . 4 ((𝐴𝐵 ∧ ∃*𝑥𝜑𝜑) → (𝑥 = 𝐴𝜓))
323expa 1203 . . 3 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝑥 = 𝐴𝜓))
43biimprd 158 . 2 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ 𝜑) → (𝜓𝑥 = 𝐴))
54impr 379 1 (((𝐴𝐵 ∧ ∃*𝑥𝜑) ∧ (𝜑𝜓)) → 𝑥 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  ∃*wmo 2027  wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741
This theorem is referenced by:  fsum3  11398  fprodseq  11594  txcn  13963
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