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Theorem sbeqalbi 38918
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 1995 . . 3 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
21alrimiv 1895 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
3 sbeqal1 38915 . 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 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by: (None)
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