Theorem wl-sb8et 37555

Description: Substitution of variable in universal quantifier. Closed form of sb8e 2522. (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 1854	. . . . 5 ⊢ (Ⅎ𝑦𝜑 ↔ Ⅎ𝑦 ¬ 𝜑)
2	1	albii 1818	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 ↔ ∀𝑥Ⅎ𝑦 ¬ 𝜑)
3		wl-sb8t 37554	. . . 4 ⊢ (∀𝑥Ⅎ𝑦 ¬ 𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
4	2, 3	sylbi 217	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥 ¬ 𝜑 ↔ ∀𝑦[𝑦 / 𝑥] ¬ 𝜑))
5		alnex 1780	. . 3 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
6		sbn 2279	. . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
7	6	albii 1818	. . . 4 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑦 / 𝑥]𝜑)
8		alnex 1780	. . . 4 ⊢ (∀𝑦 ¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
9	7, 8	bitri 275	. . 3 ⊢ (∀𝑦[𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑)
10	4, 5, 9	3bitr3g 313	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (¬ ∃𝑥𝜑 ↔ ¬ ∃𝑦[𝑦 / 𝑥]𝜑))
11	10	con4bid 317	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1537  ∃wex 1778  Ⅎwnf 1782  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-sb 2064

This theorem is referenced by:  wl-sb8mot  37582