Theorem sb8f 2359

Description: Substitution of variable in universal quantifier. Version of sb8 2525 with a disjoint variable condition, not requiring ax-10 2141 or ax-13 2380. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2141. (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 2085	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	albii 1817	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
3		alcom 2160	. 2 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑))
4		sb6 2085	. . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑))
5		sb8f.nf	. . . . 5 ⊢ Ⅎ𝑦𝜑
6	5	sbf 2272	. . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ 𝜑)
7		equcom 2017	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	imbi1i 349	. . . . 5 ⊢ ((𝑦 = 𝑥 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
9	8	albii 1817	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
10	4, 6, 9	3bitr3ri 302	. . 3 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
11	10	albii 1817	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥𝜑)
12	2, 3, 11	3bitrri 298	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  Ⅎwnf 1781  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by:  mo5f  32517  ax11-pm2  36802  bj-nfcf  36889