Theorem sb8v 2358

Description: Substitution of variable in universal quantifier. Version of sb8f 2359 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2178. (Contributed by SN, 5-Dec-2024.)

Assertion

Ref	Expression
sb8v	⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝜑,𝑦   𝑥,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sb8v

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		equcom 2017	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
5	4	imbi1i 349	. . . . 5 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑦 = 𝑥 → 𝜑))
6	5	albii 1817	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑))
7		equsv 2002	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑)
8	6, 7	bitri 275	. . 3 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
9	8	albii 1817	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥𝜑)
10	2, 3, 9	3bitrri 298	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  [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

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

This theorem is referenced by:  sbnf2  2364  sb8eulem  2601  cbvralsvw  3323  abv  3500  abvALT  3501  wl-sb8eutv  37533  sbcalf  38074