Theorem wl-sbnf1 37578

Description: Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2361. Note: This theorem shows that sbnf2 2361 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 2283	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nfa1 2152	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
3		wl-sbhbt 37577	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
4	2, 3	albid 2223	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
5	1, 4	bitrid 283	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (Ⅎ𝑥𝜑 ↔ ∀𝑥∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  Ⅎwnf 1783  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-sb 2066

This theorem is referenced by: (None)