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Theorem frege56b 43602
Description: Lemma for frege57b 43603. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege56b ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))

Proof of Theorem frege56b
StepHypRef Expression
1 frege55b 43601 . 2 (𝑦 = 𝑥𝑥 = 𝑦)
2 frege9 43516 . 2 ((𝑦 = 𝑥𝑥 = 𝑦) → ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))))
31, 2ax-mp 5 1 ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-13 2366  ax-ext 2697  ax-frege1 43494  ax-frege2 43495  ax-frege8 43513  ax-frege52c 43592
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sbc 3776
This theorem is referenced by:  frege57b  43603
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