Theorem wl-sb8t 37836

Description: Substitution of variable in universal quantifier. Closed form of sb8 2522. (Contributed by Wolf Lammen, 27-Jul-2019.)

Assertion

Ref	Expression
wl-sb8t	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

Proof of Theorem wl-sb8t

Step	Hyp	Ref	Expression
1		nfa1 2157	. 2 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
2		nfnf1 2160	. . 3 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
3	2	nfal 2329	. 2 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
4		sp 2191	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
5		wl-nfs1t 37821	. . 3 ⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
6	5	sps 2193	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
7		sbequ12 2259	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8	7	a1i 11	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)))
9	1, 3, 4, 6, 8	cbv2 2408	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  Ⅎwnf 1785  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-sb 2069

This theorem is referenced by:  wl-sb8et  37837  wl-sbhbt  37838