Theorem frege66b 43873

Description: Swap antecedents of frege65b 43872. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege66b	⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)))

Proof of Theorem frege66b

Step	Hyp	Ref	Expression
1		frege65b 43872	. 2 ⊢ (∀𝑥(𝜒 → 𝜑) → (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)))
2		ax-frege8 43771	. 2 ⊢ ((∀𝑥(𝜒 → 𝜑) → (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓))) → (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓))))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)))

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2178  ax-frege1 43752  ax-frege2 43753  ax-frege8 43771  ax-frege58b 43863

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by: (None)