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Theorem frege60c 40541
Description: Swap antecedents of frege58c 40539. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege60c (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))

Proof of Theorem frege60c
StepHypRef Expression
1 frege59c.a . . . 4 𝐴𝐵
21frege58c 40539 . . 3 (∀𝑥(𝜑 → (𝜓𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓𝜒)))
3 sbcim1 3810 . . . 4 ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)))
4 sbcim1 3810 . . . 4 ([𝐴 / 𝑥](𝜓𝜒) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
53, 4syl6 35 . . 3 ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
62, 5syl 17 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒)))
7 frege12 40431 . 2 ((∀𝑥(𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))) → (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒))))
86, 7ax-mp 5 1 (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  wcel 2115  [wsbc 3758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796  ax-frege1 40408  ax-frege2 40409  ax-frege8 40427  ax-frege58b 40519
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-sbc 3759
This theorem is referenced by:  frege93  40574
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