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Theorem sbeqalb 3455
 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 340 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝜑𝑥 = 𝐵) ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
21biimpa 500 . . . 4 (((𝜑𝑥 = 𝐴) ∧ (𝜑𝑥 = 𝐵)) → (𝑥 = 𝐴𝑥 = 𝐵))
32biimpd 218 . . 3 (((𝜑𝑥 = 𝐴) ∧ (𝜑𝑥 = 𝐵)) → (𝑥 = 𝐴𝑥 = 𝐵))
43alanimi 1734 . 2 ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → ∀𝑥(𝑥 = 𝐴𝑥 = 𝐵))
5 sbceqal 3454 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
64, 5syl5 33 1 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by:  iotaval  5779
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