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Theorem frege57c 43381
Description: Swap order of implication in ax-frege52c 43349. 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 43349 . 2 (𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 frege57c.a . . 3 𝐴𝐶
32frege56c 43380 . 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 1533  wcel 2098  [wsbc 3778
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 2699  ax-frege1 43251  ax-frege2 43252  ax-frege8 43270  ax-frege52c 43349
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-sbc 3779  df-sn 4633
This theorem is referenced by: (None)
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