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Theorem frege53b 40445
Description: Lemma for frege102 (via frege92 40510). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege53b ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))

Proof of Theorem frege53b
StepHypRef Expression
1 frege52b 40444 . 2 (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑))
2 ax-frege8 40364 . 2 ((𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑)))
31, 2ax-mp 5 1 ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796  ax-frege8 40364  ax-frege52c 40443
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-clab 2803  df-cleq 2817  df-clel 2896  df-sbc 3759
This theorem is referenced by:  frege55lem2b  40451
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