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Theorem frege53b 37003
Description: Lemma for frege102 (via frege92 37068). 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 37002 . 2 (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜑))
2 ax-frege8 36922 . 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 1865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-ext 2585  ax-frege8 36922  ax-frege52c 37001
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-clab 2592  df-cleq 2598  df-clel 2601  df-sbc 3398
This theorem is referenced by:  frege55lem2b  37009
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