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Theorem frege52b 39023
 Description: The case when the content of 𝑥 is identical with the content of 𝑦 and in which a proposition controlled by an element for which we substitute the content of 𝑥 is affirmed and the same proposition, this time where we substitute the content of 𝑦, is denied does not take place. In [𝑥 / 𝑧]𝜑, 𝑥 can also occur in other than the argument (𝑧) places. Hence 𝑥 may still be contained in [𝑦 / 𝑧]𝜑. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege52b (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Proof of Theorem frege52b
StepHypRef Expression
1 ax-frege52c 39022 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑[𝑦 / 𝑧]𝜑))
2 sbsbc 3666 . 2 ([𝑥 / 𝑧]𝜑[𝑥 / 𝑧]𝜑)
3 sbsbc 3666 . 2 ([𝑦 / 𝑧]𝜑[𝑦 / 𝑧]𝜑)
41, 2, 33imtr4g 288 1 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 2069  [wsbc 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-ext 2803  ax-frege52c 39022 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1881  df-clab 2812  df-cleq 2818  df-clel 2821  df-sbc 3663 This theorem is referenced by:  frege53b  39024  frege57b  39033
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