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Theorem sbceqal 3857
Description: Class version of one implication of equvelv 2028. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by SN, 26-Oct-2024.)
Assertion
Ref Expression
sbceqal (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sbceqal
StepHypRef Expression
1 eqeq1 2739 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐴𝐴 = 𝐴))
2 eqeq1 2739 . . . 4 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
31, 2imbi12d 344 . . 3 (𝑥 = 𝐴 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ (𝐴 = 𝐴𝐴 = 𝐵)))
4 eqid 2735 . . . 4 𝐴 = 𝐴
54a1bi 362 . . 3 (𝐴 = 𝐵 ↔ (𝐴 = 𝐴𝐴 = 𝐵))
63, 5bitr4di 289 . 2 (𝑥 = 𝐴 → ((𝑥 = 𝐴𝑥 = 𝐵) ↔ 𝐴 = 𝐵))
76spcgv 3596 1 (𝐴𝑉 → (∀𝑥(𝑥 = 𝐴𝑥 = 𝐵) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535   = wceq 1537  wcel 2106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480
This theorem is referenced by:  sbeqalb  3859
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