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Theorem frege55lem1b 38506
 Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
Assertion
Ref Expression
frege55lem1b ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑𝑥 = 𝑧))
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem frege55lem1b
StepHypRef Expression
1 equsb3 2460 . . 3 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
21biimpi 206 . 2 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
32imim2i 16 1 ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑𝑥 = 𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 1937 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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