Theorem wl-sbnf1 35941

Description: Two ways expressing that 𝑥 is effectively not free in 𝜑. Simplified version of sbnf2 2356. Note: This theorem shows that sbnf2 2356 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 2280	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
2		nfa1 2149	. . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
3		wl-sbhbt 35940	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → ((𝜑 → ∀𝑥𝜑) ↔ ∀𝑦(𝜑 → [𝑦 / 𝑥]𝜑)))
4	2, 3	albid 2216	. 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 205  ∀wal 1540  Ⅎwnf 1786  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-11 2155  ax-12 2172  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069

This theorem is referenced by: (None)