Theorem frege58bcor 44657

Description: Lemma for frege59b 44658. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege58bcor	⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem frege58bcor

Step	Hyp	Ref	Expression
1		ax-frege58b 44655	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
2		sbim 2337	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
3	1, 2	sylib 221	1 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212  ax-frege58b 44655

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813  df-sb 2096

This theorem is used by:  frege59b  44658  frege62b  44661