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Theorem sbeqalbi 39127
 Description: When both 𝑥 and 𝑧 and 𝑦 and 𝑧 are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)
Assertion
Ref Expression
sbeqalbi (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Distinct variable groups:   𝑦,𝑧   𝑥,𝑧

Proof of Theorem sbeqalbi
StepHypRef Expression
1 equtrr 2107 . . 3 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
21alrimiv 2007 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
3 sbeqal1 39124 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
42, 3impbii 199 1 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203  ax-13 2408 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858  df-sb 2050 This theorem is referenced by: (None)
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