Theorem frege67b 41050

Description: Lemma for frege68b 41051. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege67b	⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))

Proof of Theorem frege67b

Step	Hyp	Ref	Expression
1		ax-frege58b 41039	. 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
2		frege7 40946	. 2 ⊢ ((∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) → (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))))
3	1, 2	ax-mp 5	1 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1540  [wsb 2073

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40928  ax-frege2 40929  ax-frege58b 41039

This theorem is referenced by:  frege68b  41051