Theorem frege60b 40258

Description: Swap antecedents of ax-frege58b 40254. 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 40254	. . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → [𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)))
2		sbim 2311	. . . 4 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)))
3		sbim 2311	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜓 → 𝜒) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
4	3	imbi2i 338	. . . 4 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
5	2, 4	bitri 277	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
6	1, 5	sylib 220	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
7		frege12 40166	. 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)))

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-frege1 40143  ax-frege2 40144  ax-frege8 40162  ax-frege58b 40254

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)