Theorem frege64b 38203

Description: Lemma for frege65b 38204. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege64b	⊢ (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓 → 𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))

Proof of Theorem frege64b

Step	Hyp	Ref	Expression
1		frege62b 38201	. 2 ⊢ ([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓 → 𝜒) → [𝑧 / 𝑦]𝜒))
2		frege18 38112	. 2 ⊢ (([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓 → 𝜒) → [𝑧 / 𝑦]𝜒)) → (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓 → 𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒))))
3	1, 2	ax-mp 5	1 ⊢ (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓 → 𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))

Syntax hints:   → wi 4  ∀wal 1481  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege58b 38195

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  frege65b  38204