Theorem sbralie 3340

Description: Implicit to explicit substitution that swaps variables in a restrictedly universally quantified expression. (Contributed by NM, 5-Sep-2004.) Avoid ax-ext 2734, df-cleq 2754, df-clel 2837. (Revised by Wolf Lammen, 10-Mar-2025.) Avoid ax-10 2175, ax-12 2212. (Revised by SN, 13-Nov-2025.)

Hypothesis

Ref	Expression
sbralie.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbralie	⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

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

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem sbralie

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 3077	. 2 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝜑))
2		elequ2 2157	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦))
3	2	imbi1d 343	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑥 ∈ 𝑧 → 𝜑) ↔ (𝑥 ∈ 𝑦 → 𝜑)))
4	3	sbievw 2127	. . 3 ⊢ ([𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ (𝑥 ∈ 𝑦 → 𝜑))
5	4	albii 1839	. 2 ⊢ (∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝜑))
6		elequ1 2149	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
7		sbralie.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
8	6, 7	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑧 → 𝜑) ↔ (𝑦 ∈ 𝑧 → 𝜓)))
9	8	cbvalvw 2056	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝜓))
10		df-ral 3077	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝜓))
11	10	sbbii 2109	. . . . . 6 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 → 𝜓))
12		sbal 2203	. . . . . 6 ⊢ ([𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 → 𝜓) ↔ ∀𝑦[𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓))
13		elequ2 2157	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧))
14	13	imbi1d 343	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝑦 ∈ 𝑥 → 𝜓) ↔ (𝑦 ∈ 𝑧 → 𝜓)))
15	14	sbievw 2127	. . . . . . 7 ⊢ ([𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓) ↔ (𝑦 ∈ 𝑧 → 𝜓))
16	15	albii 1839	. . . . . 6 ⊢ (∀𝑦[𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝜓))
17	11, 12, 16	3bitrri 300	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑧 → 𝜓) ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
18	9, 17	bitri 277	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
19	18	sbbii 2109	. . 3 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑦 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
20		sbal 2203	. . 3 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑))
21		sbco2vv 2133	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
22	19, 20, 21	3bitr3i 303	. 2 ⊢ (∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
23	1, 5, 22	3bitr2i 301	1 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558  [wsb 2090  ∀wral 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-sb 2091  df-ral 3077

This theorem is referenced by:  tfinds2  7844