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Theorem frege58c 43911
Description: Principle related to sp 2181. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege58c.a 𝐴𝐵
Assertion
Ref Expression
frege58c (∀𝑥𝜑[𝐴 / 𝑥]𝜑)

Proof of Theorem frege58c
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 frege58c.a . 2 𝐴𝐵
2 ax-frege58b 43891 . . . . 5 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3 sbsbc 3795 . . . . 5 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3sylib 218 . . . 4 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
5 dfsbcq 3793 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
64, 5imbitrid 244 . . 3 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
76vtocleg 3553 . 2 (𝐴𝐵 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
81, 7ax-mp 5 1 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1535   = wceq 1537  [wsb 2062  wcel 2106  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-frege58b 43891
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792
This theorem is referenced by:  frege59c  43912  frege60c  43913  frege61c  43914  frege62c  43915  frege67c  43920  frege72  43925  frege118  43971  frege120  43973
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