Theorem frege60b 37219

Description: Swap antecedents of ax-frege58b 37215. 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 37215	. . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → [𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)))
2		sbim 2383	. . . 4 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)))
3		sbim 2383	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜓 → 𝜒) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
4	3	imbi2i 325	. . . 4 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
5	2, 4	bitri 263	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
6	1, 5	sylib 207	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
7		frege12 37127	. 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 1473  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege58b 37215

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by: (None)