Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege56c Structured version   Visualization version   GIF version

Theorem frege56c 41527
Description: Lemma for frege57c 41528. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege56c.b 𝐵𝐶
Assertion
Ref Expression
frege56c ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem frege56c
StepHypRef Expression
1 frege56c.b . . . . 5 𝐵𝐶
21frege54cor1c 41523 . . . 4 [𝐵 / 𝑥]𝑥 = 𝐵
3 frege53c 41522 . . . 4 ([𝐵 / 𝑥]𝑥 = 𝐵 → (𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵))
42, 3ax-mp 5 . . 3 (𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵)
5 frege55lem1c 41524 . . 3 ((𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴𝐴 = 𝐵))
64, 5ax-mp 5 . 2 (𝐵 = 𝐴𝐴 = 𝐵)
7 frege9 41420 . 2 ((𝐵 = 𝐴𝐴 = 𝐵) → ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))))
86, 7ax-mp 5 1 ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  [wsbc 3716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-frege1 41398  ax-frege2 41399  ax-frege8 41417  ax-frege52c 41496
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-sbc 3717  df-sn 4562
This theorem is referenced by:  frege57c  41528
  Copyright terms: Public domain W3C validator