Theorem frege64b 44490

Description: Lemma for frege65b 44491. 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 44488	. 2 ⊢ ([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓 → 𝜒) → [𝑧 / 𝑦]𝜒))
2		frege18 44399	. 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 1560  [wsb 2092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-10 2177  ax-12 2214  ax-frege1 44371  ax-frege2 44372  ax-frege8 44390  ax-frege58b 44482

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-nf 1806  df-sb 2093

This theorem is referenced by:  frege65b  44491