Theorem sb8v 2350

Description: Substitution of variable in universal quantifier. Version of sb8f 2351 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2171. (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 2088	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
2	1	albii 1822	. 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
3		alcom 2156	. 2 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑))
4		equcom 2021	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
5	4	imbi1i 350	. . . . 5 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑦 = 𝑥 → 𝜑))
6	5	albii 1822	. . . 4 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑))
7		equsv 2006	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑)
8	6, 7	bitri 274	. . 3 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)
9	8	albii 1822	. 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥𝜑)
10	2, 3, 9	3bitrri 298	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-11 2154

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068

This theorem is referenced by:  sbnf2  2356  sb8eulem  2598  abv  3443  abvALT  3444  sbcalf  36272