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Theorem frege55b 37211
 Description: Lemma for frege57b 37213. Proposition 55 of [Frege1879] p. 50. Note that eqtr2 2630 incorporates eqcom 2617 which is stronger than this proposition which is identical to equcomi 1931. Is is possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege55b (𝑥 = 𝑦𝑦 = 𝑥)

Proof of Theorem frege55b
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 frege55lem2b 37210 . 2 (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
2 df-sb 1868 . . 3 ([𝑦 / 𝑧]𝑧 = 𝑥 ↔ ((𝑧 = 𝑦𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦𝑧 = 𝑥)))
3 eqtr2 2630 . . . . 5 ((𝑧 = 𝑦𝑧 = 𝑥) → 𝑦 = 𝑥)
43exlimiv 1845 . . . 4 (∃𝑧(𝑧 = 𝑦𝑧 = 𝑥) → 𝑦 = 𝑥)
54adantl 481 . . 3 (((𝑧 = 𝑦𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑦 = 𝑥)
62, 5sylbi 206 . 2 ([𝑦 / 𝑧]𝑧 = 𝑥𝑦 = 𝑥)
71, 6syl 17 1 (𝑥 = 𝑦𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234  ax-ext 2590  ax-frege8 37123  ax-frege52c 37202 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-sbc 3403 This theorem is referenced by:  frege56b  37212
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