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Theorem frege55c 43245
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 3472 . . . 4 𝑥 ∈ V
21frege54cor1c 43242 . . 3 [𝑥 / 𝑦]𝑦 = 𝑥
3 frege53c 43241 . . 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 3773 . . . 4 ([𝐴 / 𝑦]𝑦 = 𝑥𝐴 ∈ {𝑦𝑦 = 𝑥})
6 clelab 2873 . . . 4 (𝐴 ∈ {𝑦𝑦 = 𝑥} ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝑥))
75, 6bitri 275 . . 3 ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝑥))
8 eqtr2 2750 . . . 4 ((𝑦 = 𝐴𝑦 = 𝑥) → 𝐴 = 𝑥)
98exlimiv 1925 . . 3 (∃𝑦(𝑦 = 𝐴𝑦 = 𝑥) → 𝐴 = 𝑥)
107, 9sylbi 216 . 2 ([𝐴 / 𝑦]𝑦 = 𝑥𝐴 = 𝑥)
114, 10syl 17 1 (𝑥 = 𝐴𝐴 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wcel 2098  {cab 2703  Vcvv 3468  [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-10 2129  ax-12 2163  ax-ext 2697  ax-frege8 43136  ax-frege52c 43215
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-sbc 3773  df-sn 4624
This theorem is referenced by:  frege104  43294
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