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Theorem frege55lem2c 37030
 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 3171 . . 3 𝑥 ∈ V
21frege54cor1c 37028 . 2 [𝑥 / 𝑧]𝑧 = 𝑥
3 frege53c 37027 . 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 1474  Vcvv 3168  [wsbc 3397 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-frege8 36922  ax-frege52c 37001 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-v 3170  df-sbc 3398  df-sn 4121 This theorem is referenced by: (None)
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