Theorem ichnfim 48273

Description: If in an interchangeability context 𝑥 is not free in 𝜑, the same holds for 𝑦. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)

Assertion

Ref	Expression
ichnfim	⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem ichnfim

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnf1 2192	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
2	1	nfal 2358	. . 3 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜑
3		nfich1 48256	. . 3 ⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑
4	2, 3	nfan 1932	. 2 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑)
5		dfich2 48267	. . . . 5 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
6		ichnfimlem 48272	. . . . . . . 8 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
7		ichnfimlem 48272	. . . . . . . 8 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑏 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑))
8	6, 7	bibi12d 348	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥𝜑 → (([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) ↔ ([𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑)))
9		bicom1 224	. . . . . . 7 ⊢ (([𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
10	8, 9	biimtrdi 256	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝜑 → (([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
11	10	2alimdv 1951	. . . . 5 ⊢ (∀𝑦Ⅎ𝑥𝜑 → (∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ∀𝑎∀𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
12	5, 11	biimtrid 245	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑥⇄𝑦]𝜑 → ∀𝑎∀𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
13	12	imp 412	. . 3 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑎∀𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
14		sbnf2 2392	. . 3 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
15	13, 14	sylibr 237	. 2 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → Ⅎ𝑦𝜑)
16	4, 15	alrimi 2252	1 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  Ⅎwnf 1816  [wsb 2099  [wich 48254

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-ich 48255

This theorem is used by:  ichnfb  48274