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Theorem sbeqalb 3732
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 344 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝜑𝑥 = 𝐵) ↔ (𝑥 = 𝐴𝑥 = 𝐵)))
21biimpa 469 . . . 4 (((𝜑𝑥 = 𝐴) ∧ (𝜑𝑥 = 𝐵)) → (𝑥 = 𝐴𝑥 = 𝐵))
32biimpd 221 . . 3 (((𝜑𝑥 = 𝐴) ∧ (𝜑𝑥 = 𝐵)) → (𝑥 = 𝐴𝑥 = 𝐵))
43alanimi 1780 . 2 ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → ∀𝑥(𝑥 = 𝐴𝑥 = 𝐵))
5 sbceqal 3731 . 2 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
64, 5syl5 34 1 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ ∀𝑥(𝜑𝑥 = 𝐵)) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1506   = wceq 1508  wcel 2051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-sbc 3675
This theorem is referenced by:  iotaval  6160
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