Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege55lem1b Structured version   Visualization version   GIF version

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)
 Copyright terms: Public domain W3C validator