Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege58c Structured version   Visualization version   GIF version

Theorem frege58c 43248
Description: Principle related to sp 2168. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege58c.a 𝐴𝐵
Assertion
Ref Expression
frege58c (∀𝑥𝜑[𝐴 / 𝑥]𝜑)

Proof of Theorem frege58c
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 frege58c.a . 2 𝐴𝐵
2 ax-frege58b 43228 . . . . 5 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3 sbsbc 3776 . . . . 5 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3sylib 217 . . . 4 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
5 dfsbcq 3774 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
64, 5imbitrid 243 . . 3 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
76vtocleg 3536 . 2 (𝐴𝐵 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
81, 7ax-mp 5 1 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531   = wceq 1533  [wsb 2059  wcel 2098  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-frege58b 43228
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-sbc 3773
This theorem is referenced by:  frege59c  43249  frege60c  43250  frege61c  43251  frege62c  43252  frege67c  43257  frege72  43262  frege118  43308  frege120  43310
  Copyright terms: Public domain W3C validator