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Theorem frege55lem2c 40633
 Description: Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege55lem2c (𝑥 = 𝐴[𝐴 / 𝑧]𝑧 = 𝑥)
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem frege55lem2c
StepHypRef Expression
1 vex 3444 . . 3 𝑥 ∈ V
21frege54cor1c 40631 . 2 [𝑥 / 𝑧]𝑧 = 𝑥
3 frege53c 40630 . 2 ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝐴[𝐴 / 𝑧]𝑧 = 𝑥))
42, 3ax-mp 5 1 (𝑥 = 𝐴[𝐴 / 𝑧]𝑧 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Vcvv 3441  [wsbc 3720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-frege8 40525  ax-frege52c 40604 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-sbc 3721  df-sn 4526 This theorem is referenced by: (None)
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