Theorem wl-sb8t 38268

Description: Substitution of variable in universal quantifier. Closed form of sb8 2551. (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 2189	. 2 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
2		nfnf1 2192	. . 3 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
3	2	nfal 2358	. 2 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
4		sp 2222	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
5		wl-nfs1t 38253	. . 3 ⊢ (Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
6	5	sps 2224	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
7		sbequ12 2289	. . 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 2437	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  wl-sb8et  38269  wl-sbhbt  38270