Theorem wl-sb8et 38267

Description: Substitution of variable in universal quantifier. Closed form of sb8e 2552. (Contributed by Wolf Lammen, 27-Jul-2019.)

Assertion

Ref	Expression
wl-sb8et	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))

Proof of Theorem wl-sb8et

Step	Hyp	Ref	Expression
1		nfnbi 1888	. . . . 5 ⊢ (Ⅎ𝑦𝜑 ↔ Ⅎ𝑦 ¬ 𝜑)
2	1	albii 1852	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥Ⅎ𝑦 ¬ 𝜑)
3		wl-sb8t 38266	. . . 4 ⊢ (∀𝑥Ⅎ𝑦 ¬ 𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
4	2, 3	sylbi 220	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
5		alnex 1814	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6		sbn 2317	. . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
7	6	albii 1852	. . . 4 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
8		alnex 1814	. . . 4 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
9	7, 8	bitri 278	. . 3 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
10	4, 5, 9	3bitr3g 316	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑))
11	10	con4bid 320	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  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-sb8mot  38294