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Theorem frege58c 39055
Description: Principle related to sp 2226. 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 39035 . . . . 5 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3 sbsbc 3666 . . . . 5 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3sylib 210 . . . 4 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
5 dfsbcq 3664 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
64, 5syl5ib 236 . . 3 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
76vtocleg 3496 . 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 1656   = wceq 1658  [wsb 2069  wcel 2166  [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-12 2222  ax-ext 2803  ax-frege58b 39035
This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416  df-sbc 3663
This theorem is referenced by:  frege59c  39056  frege60c  39057  frege61c  39058  frege62c  39059  frege67c  39064  frege72  39069  frege118  39115  frege120  39117
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