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Theorem frege58c 42267
Description: Principle related to sp 2177. 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 42247 . . . . 5 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3 sbsbc 3748 . . . . 5 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3sylib 217 . . . 4 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
5 dfsbcq 3746 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
64, 5imbitrid 243 . . 3 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
76vtocleg 3517 . 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 1540   = wceq 1542  [wsb 2068  wcel 2107  [wsbc 3744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-frege58b 42247
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-sbc 3745
This theorem is referenced by:  frege59c  42268  frege60c  42269  frege61c  42270  frege62c  42271  frege67c  42276  frege72  42281  frege118  42327  frege120  42329
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