Theorem frege60b 44659

Description: Swap antecedents of ax-frege58b 44655. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege60b	⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

Proof of Theorem frege60b

Step	Hyp	Ref	Expression
1		ax-frege58b 44655	. . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → [𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)))
2		sbim 2337	. . . 4 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)))
3		sbim 2337	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜓 → 𝜒) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
4	3	imbi2i 339	. . . 4 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
5	2, 4	bitri 278	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
6	1, 5	sylib 221	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
7		frege12 44567	. 2 ⊢ ((∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))) → (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))))
8	6, 7	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212  ax-frege1 44544  ax-frege2 44545  ax-frege8 44563  ax-frege58b 44655

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813  df-sb 2096

This theorem is used by: (None)