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Theorem sbeqalb 3019
Description: Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.)
Assertion
Ref Expression
sbeqalb (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → 𝐴 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbeqalb
StepHypRef Expression
1 bibi1 240 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝜑𝑥 = 𝐵) ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
21biimpa 296 . . . 4 (((𝜑𝑥 = 𝐴) ∧ (𝜑𝑥 = 𝐵)) → (𝑥 = 𝐴𝑥 = 𝐵))
32biimpd 144 . . 3 (((𝜑𝑥 = 𝐴) ∧ (𝜑𝑥 = 𝐵)) → (𝑥 = 𝐴𝑥 = 𝐵))
43alanimi 1459 . 2 ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → ∀𝑥(𝑥 = 𝐴𝑥 = 𝐵))
5 sbceqal 3018 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
64, 5syl5 32 1 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → 𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  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-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963
This theorem is referenced by:  iotaval  5188
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