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Theorem frege55c 40536
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 3483 . . . 4 𝑥 ∈ V
21frege54cor1c 40533 . . 3 [𝑥 / 𝑦]𝑦 = 𝑥
3 frege53c 40532 . . 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 3759 . . . 4 ([𝐴 / 𝑦]𝑦 = 𝑥𝐴 ∈ {𝑦𝑦 = 𝑥})
6 clelab 2959 . . . 4 (𝐴 ∈ {𝑦𝑦 = 𝑥} ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝑥))
75, 6bitri 278 . . 3 ([𝐴 / 𝑦]𝑦 = 𝑥 ↔ ∃𝑦(𝑦 = 𝐴𝑦 = 𝑥))
8 eqtr2 2845 . . . 4 ((𝑦 = 𝐴𝑦 = 𝑥) → 𝐴 = 𝑥)
98exlimiv 1932 . . 3 (∃𝑦(𝑦 = 𝐴𝑦 = 𝑥) → 𝐴 = 𝑥)
107, 9sylbi 220 . 2 ([𝐴 / 𝑦]𝑦 = 𝑥𝐴 = 𝑥)
114, 10syl 17 1 (𝑥 = 𝐴𝐴 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2115  {cab 2802  Vcvv 3480  [wsbc 3758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-frege8 40427  ax-frege52c 40506
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-sbc 3759  df-sn 4551
This theorem is referenced by:  frege104  40585
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