Theorem cbvralsvw 3322

Description: Change bound variable by using a substitution. Version of cbvralsv 3362 with a disjoint variable condition, which does not require ax-13 2410. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2410. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) Avoid ax-10 2182, ax-12 2219. (Revised by SN, 21-Aug-2025.)

Assertion

Ref	Expression
cbvralsvw	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvralsvw

Step	Hyp	Ref	Expression
1		sb8v 2391	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑))
2		df-ral 3086	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		df-ral 3086	. . 3 ⊢ (∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑))
4		eleq1w 2852	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	4	imbi1d 344	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜑)))
6	5	sbbiiev 2133	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑) ↔ [𝑦 / 𝑥](𝑦 ∈ 𝐴 → 𝜑))
7		sbrimvw 2131	. . . . 5 ⊢ ([𝑦 / 𝑥](𝑦 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑))
8	6, 7	bitr2i 279	. . . 4 ⊢ ((𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑) ↔ [𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑))
9	8	albii 1846	. . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑦[𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑))
10	3, 9	bitri 278	. 2 ⊢ (∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦[𝑦 / 𝑥](𝑥 ∈ 𝐴 → 𝜑))
11	1, 2, 10	3bitr4i 306	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  [wsb 2097   ∈ wcel 2149  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-11 2198

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clel 2844  df-ral 3086

This theorem is referenced by:  sbralieALT  3350  rspsbc  3841  ralxpf  5830  tfinds  7852  tfindes  7855  nn0min  33102