Theorem ichnfimlem 44542

Description: Lemma for ichnfim 44543: A substitution for a nonfree variable has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) Avoid ax-13 2369. (Revised by Gino Giotto, 1-May-2024.)

Assertion

Ref	Expression
ichnfimlem	⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))

Distinct variable group:   𝑥,𝑏,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem ichnfimlem

Step	Hyp	Ref	Expression
1		nfa1 2152	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑥𝜑
2		sb6 2091	. . . . . . . . . 10 ⊢ ([𝑏 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑏 → 𝜑))
3	2	a1i 11	. . . . . . . . 9 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑏 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑏 → 𝜑)))
4	2	biimpri 231	. . . . . . . . . 10 ⊢ (∀𝑦(𝑦 = 𝑏 → 𝜑) → [𝑏 / 𝑦]𝜑)
5	4	axc4i 2321	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑏 → 𝜑) → ∀𝑦[𝑏 / 𝑦]𝜑)
6	3, 5	syl6bi 256	. . . . . . . 8 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑏 / 𝑦]𝜑 → ∀𝑦[𝑏 / 𝑦]𝜑))
7	1, 6	nf5d 2285	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑦[𝑏 / 𝑦]𝜑)
8	1, 7	nfim1 2197	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑)
9		sbequ12 2249	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝜑 ↔ [𝑏 / 𝑦]𝜑))
10	9	imbi2d 344	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((∀𝑦Ⅎ𝑥𝜑 → 𝜑) ↔ (∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑)))
11	8, 10	equsalv 2264	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑏 → (∀𝑦Ⅎ𝑥𝜑 → 𝜑)) ↔ (∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑))
12	11	bicomi 227	. . . 4 ⊢ ((∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑) ↔ ∀𝑦(𝑦 = 𝑏 → (∀𝑦Ⅎ𝑥𝜑 → 𝜑)))
13		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑏
14		nfnf1 2155	. . . . . . . 8 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
15	14	nfal 2322	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜑
16		sp 2180	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
17	15, 16	nfim1 2197	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜑 → 𝜑)
18	13, 17	nfim 1904	. . . . 5 ⊢ Ⅎ𝑥(𝑦 = 𝑏 → (∀𝑦Ⅎ𝑥𝜑 → 𝜑))
19	18	nfal 2322	. . . 4 ⊢ Ⅎ𝑥∀𝑦(𝑦 = 𝑏 → (∀𝑦Ⅎ𝑥𝜑 → 𝜑))
20	12, 19	nfxfr 1860	. . 3 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑)
21		pm5.5 365	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ((∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑) ↔ [𝑏 / 𝑦]𝜑))
22	15, 21	nfbidf 2222	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → (Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑) ↔ Ⅎ𝑥[𝑏 / 𝑦]𝜑))
23	20, 22	mpbii 236	. 2 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥[𝑏 / 𝑦]𝜑)
24		sbft 2266	. 2 ⊢ (Ⅎ𝑥[𝑏 / 𝑦]𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
25	23, 24	syl 17	1 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  Ⅎwnf 1791  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-sb 2071

This theorem is referenced by:  ichnfim  44543