Theorem frege66b 39044

Description: Swap antecedents of frege65b 39043. 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 39043	. 2 ⊢ (∀𝑥(𝜒 → 𝜑) → (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)))
2		ax-frege8 38942	. 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 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)