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Theorem frege60b 40621
 Description: Swap antecedents of ax-frege58b 40617. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege60b (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

Proof of Theorem frege60b
StepHypRef Expression
1 ax-frege58b 40617 . . 3 (∀𝑥(𝜑 → (𝜓𝜒)) → [𝑦 / 𝑥](𝜑 → (𝜓𝜒)))
2 sbim 2307 . . . 4 ([𝑦 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓𝜒)))
3 sbim 2307 . . . . 5 ([𝑦 / 𝑥](𝜓𝜒) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
43imbi2i 339 . . . 4 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
52, 4bitri 278 . . 3 ([𝑦 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
61, 5sylib 221 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
7 frege12 40529 . 2 ((∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))) → (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))))
86, 7ax-mp 5 1 (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  [wsb 2069 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 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-frege1 40506  ax-frege2 40507  ax-frege8 40525  ax-frege58b 40617 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by: (None)
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