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Theorem frege52b 37692
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 subsitute 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 37691 . 2 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑[𝑦 / 𝑧]𝜑))
2 sbsbc 3425 . 2 ([𝑥 / 𝑧]𝜑[𝑥 / 𝑧]𝜑)
3 sbsbc 3425 . 2 ([𝑦 / 𝑧]𝜑[𝑦 / 𝑧]𝜑)
41, 2, 33imtr4g 285 1 (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  [wsb 1877  [wsbc 3421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-ext 2601  ax-frege52c 37691
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-clab 2608  df-cleq 2614  df-clel 2617  df-sbc 3422
This theorem is referenced by:  frege53b  37693  frege57b  37702
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