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Theorem frege55lem2b 42256
Description: Lemma for frege55b 42257. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege55lem2b (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

Proof of Theorem frege55lem2b
StepHypRef Expression
1 frege54cor1b 42254 . 2 [𝑥 / 𝑧]𝑧 = 𝑥
2 frege53b 42250 . 2 ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥))
31, 2ax-mp 5 1 (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-13 2371  ax-ext 2704  ax-frege8 42169  ax-frege52c 42248
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sbc 3741
This theorem is referenced by:  frege55b  42257
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