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Theorem frege57c 38740
Description: Swap order of implication in ax-frege52c 38708. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege57c.a 𝐴𝐶
Assertion
Ref Expression
frege57c (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem frege57c
StepHypRef Expression
1 ax-frege52c 38708 . 2 (𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 frege57c.a . . 3 𝐴𝐶
32frege56c 38739 . 2 ((𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)) → (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)))
41, 3ax-mp 5 1 (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-frege1 38610  ax-frege2 38611  ax-frege8 38629  ax-frege52c 38708
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3588  df-sn 4317
This theorem is referenced by: (None)
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