Theorem ichnfimlem 48546

Description: Lemma for ichnfim 48547: A substitution for a nonfree variable has no effect. (Contributed by Wolf Lammen, 6-Aug-2023.) Avoid ax-13 2402. (Revised by GG, 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 2188	. . . . . . 7 ⊢ Ⅎ𝑦∀𝑦Ⅎ𝑥𝜑
2		sb6 2122	. . . . . . . . . 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 2353	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 = 𝑏 → 𝜑) → ∀𝑦[𝑏 / 𝑦]𝜑)
6	3, 5	biimtrdi 256	. . . . . . . 8 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑏 / 𝑦]𝜑 → ∀𝑦[𝑏 / 𝑦]𝜑))
7	1, 6	nf5d 2318	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑦[𝑏 / 𝑦]𝜑)
8	1, 7	nfim1 2236	. . . . . 6 ⊢ Ⅎ𝑦(∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑)
9		sbequ12 2287	. . . . . . 7 ⊢ (𝑦 = 𝑏 → (𝜑 ↔ [𝑏 / 𝑦]𝜑))
10	9	imbi2d 343	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((∀𝑦Ⅎ𝑥𝜑 → 𝜑) ↔ (∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑)))
11	8, 10	equsalv 2302	. . . . 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 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 = 𝑏
14		nfnf1 2191	. . . . . . . 8 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
15	14	nfal 2354	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑦Ⅎ𝑥𝜑
16		sp 2220	. . . . . . 7 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
17	15, 16	nfim1 2236	. . . . . 6 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜑 → 𝜑)
18	13, 17	nfim 1929	. . . . 5 ⊢ Ⅎ𝑥(𝑦 = 𝑏 → (∀𝑦Ⅎ𝑥𝜑 → 𝜑))
19	18	nfal 2354	. . . 4 ⊢ Ⅎ𝑥∀𝑦(𝑦 = 𝑏 → (∀𝑦Ⅎ𝑥𝜑 → 𝜑))
20	12, 19	nfxfr 1886	. . 3 ⊢ Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑)
21		pm5.5 364	. . . 4 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ((∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑) ↔ [𝑏 / 𝑦]𝜑))
22	15, 21	nfbidf 2261	. . 3 ⊢ (∀𝑦Ⅎ𝑥𝜑 → (Ⅎ𝑥(∀𝑦Ⅎ𝑥𝜑 → [𝑏 / 𝑦]𝜑) ↔ Ⅎ𝑥[𝑏 / 𝑦]𝜑))
23	20, 22	mpbii 236	. 2 ⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥[𝑏 / 𝑦]𝜑)
24		sbft 2304	. 2 ⊢ (Ⅎ𝑥[𝑏 / 𝑦]𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))
25	23, 24	syl 18	1 ⊢ (∀𝑦Ⅎ𝑥𝜑 → ([𝑎 / 𝑥][𝑏 / 𝑦]𝜑 ↔ [𝑏 / 𝑦]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  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 2178  ax-11 2194  ax-12 2213

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:  ichnfim  48547