Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege57b Structured version   Visualization version   GIF version

Theorem frege57b 39034
Description: Analogue of frege57aid 39007. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege57b (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))

Proof of Theorem frege57b
StepHypRef Expression
1 frege52b 39024 . 2 (𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
2 frege56b 39033 . 2 ((𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)) → (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑)))
31, 2ax-mp 5 1 (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
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 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-13 2391  ax-ext 2804  ax-frege1 38925  ax-frege2 38926  ax-frege8 38944  ax-frege52c 39023
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-sbc 3664
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator