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Theorem frege55c 38529
Description: Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege55c (𝑥 = 𝐴𝐴 = 𝑥)

Proof of Theorem frege55c
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3234 . . . 4 𝑥 ∈ V
21frege54cor1c 38526 . . 3 [𝑥 / 𝑦]𝑦 = 𝑥
3 frege53c 38525 . . 3 ([𝑥 / 𝑦]𝑦 = 𝑥 → (𝑥 = 𝐴[𝐴 / 𝑦]𝑦 = 𝑥))
42, 3ax-mp 5 . 2 (𝑥 = 𝐴[𝐴 / 𝑦]𝑦 = 𝑥)
5 df-sbc 3469 . . . 4 ([𝐴 / 𝑦]𝑦 = 𝑥𝐴 ∈ {𝑦𝑦 = 𝑥})
6 clelab 2777 . . . 4 (𝐴 ∈ {𝑦𝑦 = 𝑥} ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝑥))
75, 6bitri 264 . . 3 ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝑥))
8 eqtr2 2671 . . . 4 ((𝑦 = 𝐴𝑦 = 𝑥) → 𝐴 = 𝑥)
98exlimiv 1898 . . 3 (∃𝑦(𝑦 = 𝐴𝑦 = 𝑥) → 𝐴 = 𝑥)
107, 9sylbi 207 . 2 ([𝐴 / 𝑦]𝑦 = 𝑥𝐴 = 𝑥)
114, 10syl 17 1 (𝑥 = 𝐴𝐴 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  Vcvv 3231  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-frege8 38420  ax-frege52c 38499
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-sn 4211
This theorem is referenced by:  frege104  38578
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