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Theorem frege60b 38516
Description: Swap antecedents of ax-frege58b 38512. 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 38512 . . 3 (∀𝑥(𝜑 → (𝜓𝜒)) → [𝑦 / 𝑥](𝜑 → (𝜓𝜒)))
2 sbim 2423 . . . 4 ([𝑦 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓𝜒)))
3 sbim 2423 . . . . 5 ([𝑦 / 𝑥](𝜓𝜒) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
43imbi2i 325 . . . 4 (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
52, 4bitri 264 . . 3 ([𝑦 / 𝑥](𝜑 → (𝜓𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
61, 5sylib 208 . 2 (∀𝑥(𝜑 → (𝜓𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
7 frege12 38424 . 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 1521  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087  ax-13 2282  ax-frege1 38401  ax-frege2 38402  ax-frege8 38420  ax-frege58b 38512
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by: (None)
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