Theorem ichnfim 43700

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 2157	. . . 4 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
2	1	nfal 2341	. . 3 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜑
3		nfich1 43688	. . 3 ⊢ Ⅎ𝑥[𝑥⇄𝑦]𝜑
4	2, 3	nfan 1899	. 2 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑)
5		dfich2 43694	. . . . 5 ⊢ ([𝑥⇄𝑦]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑))
6		ichnfimlem 43699	. . . . . . . 8 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
7		ichnfimlem 43699	. . . . . . . 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 223	. . . . . . 7 ⊢ (([𝑏 / 𝑦]𝜑 ↔ [𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
10	8, 9	syl6bi 255	. . . . . 6 ⊢ (∀𝑦Ⅎ𝑥𝜑 → (([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑥][𝑎 / 𝑦]𝜑) → ([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
11	10	2alimdv 1918	. . . . 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	syl5bi 244	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑥⇄𝑦]𝜑 → ∀𝑎∀𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑)))
13	12	imp 409	. . 3 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑎∀𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
14		sbnf2 2376	. . 3 ⊢ (Ⅎ𝑦𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
15	13, 14	sylibr 236	. 2 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → Ⅎ𝑦𝜑)
16	4, 15	alrimi 2212	1 ⊢ ((∀𝑦Ⅎ𝑥𝜑 ∧ [𝑥⇄𝑦]𝜑) → ∀𝑥Ⅎ𝑦𝜑)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1534  Ⅎwnf 1783  [wsb 2068  [wich 43686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-ich 43687

This theorem is referenced by:  ichnfb  43701