Theorem sb8f 2384

Description: Substitution of variable in universal quantifier. Version of sb8 2547 with a disjoint variable condition, not requiring ax-10 2174 or ax-13 2402. (Contributed by NM, 16-May-1993.) (Revised by Wolf Lammen, 19-Jan-2023.) Avoid ax-10 2174. (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 2117	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	albii 1838	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
3		alcom 2192	. 2 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑))
4		sb6 2117	. . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑))
5		sb8f.nf	. . . . 5 ⊢ Ⅎ𝑦𝜑
6	5	sbf 2304	. . . 4 ⊢ ([𝑥 / 𝑦]𝜑 ↔ 𝜑)
7		equcom 2037	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	imbi1i 351	. . . . 5 ⊢ ((𝑦 = 𝑥 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑))
9	8	albii 1838	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
10	4, 6, 9	3bitr3ri 304	. . 3 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
11	10	albii 1838	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥𝜑)
12	2, 3, 11	3bitrri 300	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1557  Ⅎwnf 1802  [wsb 2089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-11 2190  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-nf 1803  df-sb 2090

This theorem is referenced by:  mo5f  32647  ax11-pm2  37282  bj-nfcf  37369