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Theorem frege55c 37035
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 3175 . . . 4 𝑥 ∈ V
21frege54cor1c 37032 . . 3 [𝑥 / 𝑦]𝑦 = 𝑥
3 frege53c 37031 . . 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 3402 . . . 4 ([𝐴 / 𝑦]𝑦 = 𝑥𝐴 ∈ {𝑦𝑦 = 𝑥})
6 clelab 2734 . . . 4 (𝐴 ∈ {𝑦𝑦 = 𝑥} ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝑥))
75, 6bitri 262 . . 3 ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝑥))
8 eqtr2 2629 . . . 4 ((𝑦 = 𝐴𝑦 = 𝑥) → 𝐴 = 𝑥)
98exlimiv 1844 . . 3 (∃𝑦(𝑦 = 𝐴𝑦 = 𝑥) → 𝐴 = 𝑥)
107, 9sylbi 205 . 2 ([𝐴 / 𝑦]𝑦 = 𝑥𝐴 = 𝑥)
114, 10syl 17 1 (𝑥 = 𝐴𝐴 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1976  {cab 2595  Vcvv 3172  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-frege8 36926  ax-frege52c 37005
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402  df-sn 4125
This theorem is referenced by:  frege104  37084
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