Theorem frege58bcor 45095

Description: Lemma for frege59b 45096. (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 45093	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓))
2		sbim 2340	. 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 1568  [wsb 2101

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216  ax-frege58b 45093

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2102

This theorem is used by:  frege59b  45096  frege62b  45099