Theorem frege68b 45022

Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege68b	⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))

Proof of Theorem frege68b

Step	Hyp	Ref	Expression
1		frege57aid 44981	. 2 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑))
2		frege67b 45021	. 2 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 44899  ax-frege2 44900  ax-frege8 44918  ax-frege52a 44966  ax-frege58b 45010

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-tru 1572  df-fal 1582

This theorem is used by: (None)