Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege52b Structured version   Visualization version   GIF version

Theorem frege52b 40235
 Description: The case when the content of 𝑥 is identical with the content of 𝑦 and in which a proposition controlled by an element for which we substitute the content of 𝑥 is affirmed and the same proposition, this time where we substitute the content of 𝑦, is denied does not take place. In [𝑥 / 𝑧]𝜑, 𝑥 can also occur in other than the argument (𝑧) places. Hence 𝑥 may still be contained in [𝑦 / 𝑧]𝜑. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege52b (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Proof of Theorem frege52b
StepHypRef Expression
1 ax-frege52c 40234 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑[𝑦 / 𝑧]𝜑))
2 sbsbc 3775 . 2 ([𝑥 / 𝑧]𝜑[𝑥 / 𝑧]𝜑)
3 sbsbc 3775 . 2 ([𝑦 / 𝑧]𝜑[𝑦 / 𝑧]𝜑)
41, 2, 33imtr4g 298 1 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 2065  [wsbc 3771 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793  ax-frege52c 40234 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-clab 2800  df-cleq 2814  df-clel 2893  df-sbc 3772 This theorem is referenced by:  frege53b  40236  frege57b  40245
 Copyright terms: Public domain W3C validator