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Theorem frege55lem1b 38958
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 2549 . . 3 ([𝑥 / 𝑦]𝑦 = 𝑧𝑥 = 𝑧)
21biimpi 208 . 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 2064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-12 2213  ax-13 2375
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ex 1876  df-nf 1880  df-sb 2065
This theorem is referenced by: (None)
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