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Theorem frege57c 39053
 Description: Swap order of implication in ax-frege52c 39021. 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 39021 . 2 (𝐵 = 𝐴 → ([𝐵 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
2 frege57c.a . . 3 𝐴𝐶
32frege56c 39052 . 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 1656   ∈ wcel 2164  [wsbc 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803  ax-frege1 38923  ax-frege2 38924  ax-frege8 38942  ax-frege52c 39021 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-sbc 3663  df-sn 4400 This theorem is referenced by: (None)
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