Theorem wl-sblimt 33332

Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2396. (Contributed by Wolf Lammen, 26-Jul-2019.)

Assertion

Ref	Expression
wl-sblimt	⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))

Proof of Theorem wl-sblimt

Step	Hyp	Ref	Expression
1		sbim 2395	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		sbft 2379	. . 3 ⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥]𝜓 ↔ 𝜓))
3	2	imbi2d 330	. 2 ⊢ (Ⅎ𝑥𝜓 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))
4	1, 3	syl5bb 272	1 ⊢ (Ⅎ𝑥𝜓 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 196  Ⅎwnf 1708  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by: (None)