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Theorem frege56c 38733
Description: Lemma for frege57c 38734. 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 38729 . . . 4 [𝐵 / 𝑥]𝑥 = 𝐵
3 frege53c 38728 . . . 4 ([𝐵 / 𝑥]𝑥 = 𝐵 → (𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵))
42, 3ax-mp 5 . . 3 (𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵)
5 frege55lem1c 38730 . . 3 ((𝐵 = 𝐴[𝐴 / 𝑥]𝑥 = 𝐵) → (𝐵 = 𝐴𝐴 = 𝐵))
64, 5ax-mp 5 . 2 (𝐵 = 𝐴𝐴 = 𝐵)
7 frege9 38626 . 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 1632  wcel 2139  [wsbc 3576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-frege1 38604  ax-frege2 38605  ax-frege8 38623  ax-frege52c 38702
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-sn 4322
This theorem is referenced by:  frege57c  38734
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