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Theorem frege55b 39032
 Description: Lemma for frege57b 39034. Proposition 55 of [Frege1879] p. 50. Note that eqtr2 2848 incorporates eqcom 2833 which is stronger than this proposition which is identical to equcomi 2123. Is it 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 39031 . 2 (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
2 df-sb 2070 . . 3 ([𝑦 / 𝑧]𝑧 = 𝑥 ↔ ((𝑧 = 𝑦𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦𝑧 = 𝑥)))
3 eqtr2 2848 . . . . 5 ((𝑧 = 𝑦𝑧 = 𝑥) → 𝑦 = 𝑥)
43exlimiv 2031 . . . 4 (∃𝑧(𝑧 = 𝑦𝑧 = 𝑥) → 𝑦 = 𝑥)
54adantl 475 . . 3 (((𝑧 = 𝑦𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑦 = 𝑥)
62, 5sylbi 209 . 2 ([𝑦 / 𝑧]𝑧 = 𝑥𝑦 = 𝑥)
71, 6syl 17 1 (𝑥 = 𝑦𝑦 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∃wex 1880  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-13 2391  ax-ext 2804  ax-frege8 38944  ax-frege52c 39023 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-sbc 3664 This theorem is referenced by:  frege56b  39033
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