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Theorem frege55lem1b 40248
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 2109 . . 3 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
21biimpi 218 . 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 2069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070
This theorem is referenced by: (None)
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