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Theorem frege56c 37057
Description: Lemma for frege57c 37058. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege56c.b 𝐵𝐶
Assertion
Ref Expression
frege56c ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem frege56c
StepHypRef Expression
1 frege56c.b . . . . 5 𝐵𝐶
21frege54cor1c 37053 . . . 4 [𝐵 / 𝑥]𝑥 = 𝐵
3 frege53c 37052 . . . 4 ([𝐵 / 𝑥]𝑥 = 𝐵 → (𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵))
42, 3ax-mp 5 . . 3 (𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵)
5 frege55lem1c 37054 . . 3 ((𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴𝐴 = 𝐵))
64, 5ax-mp 5 . 2 (𝐵 = 𝐴𝐴 = 𝐵)
7 frege9 36950 . 2 ((𝐵 = 𝐴𝐴 = 𝐵) → ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))))
86, 7ax-mp 5 1 ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  [wsbc 3401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-frege1 36928  ax-frege2 36929  ax-frege8 36947  ax-frege52c 37026
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 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402  df-sn 4125
This theorem is referenced by:  frege57c  37058
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