Theorem sb8f 2386

Description: Substitution of variable in universal quantifier. Version of sb8 2549 with a disjoint variable condition, not requiring ax-10 2180 or ax-13 2404. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2180. (Revised by SN, 5-Dec-2024.)

Hypothesis

Ref	Expression
sb8f.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8f	⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem sb8f

Step	Hyp	Ref	Expression
1		sb6 2124	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	albii 1852	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
3		alcom 2199	. 2 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑))
4		sb6 2124	. . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑))
5		sb8f.nf	. . . . 5 ⊢ Ⅎ𝑦𝜑
6	5	sbf 2308	. . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ 𝜑)
7		equcom 2053	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	imbi1i 352	. . . . 5 ⊢ ((𝑦 = 𝑥 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
9	8	albii 1852	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
10	4, 6, 9	3bitr3ri 305	. . 3 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
11	10	albii 1852	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥𝜑)
12	2, 3, 11	3bitrri 301	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816  [wsb 2101

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 1945  ax-6 2002  ax-7 2043  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2102

This theorem is used by:  mo5f  33184  ax11-pm2  37963  bj-nfcf  38050