Theorem frege60b 39038

Description: Swap antecedents of ax-frege58b 39034. 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 39034	. . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → [𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)))
2		sbim 2526	. . . 4 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)))
3		sbim 2526	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜓 → 𝜒) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))
4	3	imbi2i 328	. . . 4 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
5	2, 4	bitri 267	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
6	1, 5	sylib 210	. 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)))
7		frege12 38946	. 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 1654  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-12 2220  ax-13 2389  ax-frege1 38923  ax-frege2 38924  ax-frege8 38942  ax-frege58b 39034

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068

This theorem is referenced by: (None)