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Theorem frege53b 38501
Description: Lemma for frege102 (via frege92 38566). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege53b ([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]𝜑))

Proof of Theorem frege53b
StepHypRef Expression
1 frege52b 38500 . 2 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜑))
2 ax-frege8 38420 . 2 ((𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜑)) → ([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]𝜑)))
31, 2ax-mp 5 1 ([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-ext 2631  ax-frege8 38420  ax-frege52c 38499
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-clab 2638  df-cleq 2644  df-clel 2647  df-sbc 3469
This theorem is referenced by:  frege55lem2b  38507
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