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Theorem equsb3lem 1950
Description: Lemma for equsb3 1951. (Contributed by NM, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
equsb3lem ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
Distinct variable groups:   𝑥,𝑧   𝑥,𝑦

Proof of Theorem equsb3lem
StepHypRef Expression
1 ax-17 1526 . 2 (𝑦 = 𝑧 → ∀𝑥 𝑦 = 𝑧)
2 equequ1 1712 . 2 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
31, 2sbieh 1790 1 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  equsb3  1951
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