Theorem frege68b 43871

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 43830	. 2 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑))
2		frege67b 43870	. 2 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 43748  ax-frege2 43749  ax-frege8 43767  ax-frege52a 43815  ax-frege58b 43859

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-tru 1542  df-fal 1552

This theorem is referenced by: (None)