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Theorem frege55lem2c 43411
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 3467 . . 3 𝑥 ∈ V
21frege54cor1c 43409 . 2 [𝑥 / 𝑧]𝑧 = 𝑥
3 frege53c 43408 . 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 1533  Vcvv 3463  [wsbc 3769
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-frege8 43303  ax-frege52c 43382
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-sbc 3770  df-sn 4625
This theorem is referenced by: (None)
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