Theorem wl-sbnf1 33336

Description: Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2439. Note: This theorem shows that sbnf2 2439 has unnecessary distinct variable constraints. (Contributed by Wolf Lammen, 28-Jul-2019.)

Assertion

Ref	Expression
wl-sbnf1	⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Proof of Theorem wl-sbnf1

Step	Hyp	Ref	Expression
1		nf5 2116	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nfa1 2028	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
3		wl-sbhbt 33335	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
4	2, 3	albid 2090	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
5	1, 4	syl5bb 272	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  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-11 2034  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)