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Theorem frege60c 37036
Description: Swap antecedents of frege58c 37034. 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 37034 . . 3 (∀𝑥(𝜑 → (𝜓𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓𝜒)))
3 sbcim1 3444 . . . 4 ([𝐴 / 𝑥](𝜑 → (𝜓𝜒)) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥](𝜓𝜒)))
4 sbcim1 3444 . . . 4 ([𝐴 / 𝑥](𝜓𝜒) → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
53, 4syl6 34 . . 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 36926 . 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 1472  wcel 1975  [wsbc 3397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-12 2031  ax-13 2228  ax-ext 2585  ax-frege1 36903  ax-frege2 36904  ax-frege8 36922  ax-frege58b 37014
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-v 3170  df-sbc 3398
This theorem is referenced by:  frege93  37069
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