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Theorem frege55b 40121
Description: Lemma for frege57b 40123. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2839 incorporates eqcom 2825 which is stronger than this proposition which is identical to equcomi 2015. 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 40120 . 2 (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
2 dfsb1 2503 . . 3 ([𝑦 / 𝑧]𝑧 = 𝑥 ↔ ((𝑧 = 𝑦𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦𝑧 = 𝑥)))
3 eqtr2 2839 . . . . 5 ((𝑧 = 𝑦𝑧 = 𝑥) → 𝑦 = 𝑥)
43exlimiv 1922 . . . 4 (∃𝑧(𝑧 = 𝑦𝑧 = 𝑥) → 𝑦 = 𝑥)
54adantl 482 . . 3 (((𝑧 = 𝑦𝑧 = 𝑥) ∧ ∃𝑧(𝑧 = 𝑦𝑧 = 𝑥)) → 𝑦 = 𝑥)
62, 5sylbi 218 . 2 ([𝑦 / 𝑧]𝑧 = 𝑥𝑦 = 𝑥)
71, 6syl 17 1 (𝑥 = 𝑦𝑦 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1771  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-13 2381  ax-ext 2790  ax-frege8 40033  ax-frege52c 40112
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-sbc 3770
This theorem is referenced by:  frege56b  40122
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