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Theorem frege58c 40151
 Description: Principle related to sp 2174. 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 40131 . . . . 5 (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
3 sbsbc 3780 . . . . 5 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
42, 3sylib 219 . . . 4 (∀𝑥𝜑[𝑦 / 𝑥]𝜑)
5 dfsbcq 3778 . . . 4 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
64, 5syl5ib 245 . . 3 (𝑦 = 𝐴 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
76vtocleg 3586 . 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 1528   = wceq 1530  [wsb 2062   ∈ wcel 2107  [wsbc 3776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798  ax-frege58b 40131 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-clab 2805  df-cleq 2819  df-clel 2898  df-sbc 3777 This theorem is referenced by:  frege59c  40152  frege60c  40153  frege61c  40154  frege62c  40155  frege67c  40160  frege72  40165  frege118  40211  frege120  40213
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