Theorem sbralie 3354

Description: Implicit to explicit substitution that swaps variables in a rectrictedly universally quantified expression. (Contributed by NM, 5-Sep-2004.) Avoid ax-ext 2703. (Revised by Wolf Lammen, 10-Mar-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 3062	. 2 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝜑))
2		sbim 2299	. . . 4 ⊢ ([𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ ([𝑦 / 𝑧]𝑥 ∈ 𝑧 → [𝑦 / 𝑧]𝜑))
3		elsb2 2123	. . . . 5 ⊢ ([𝑦 / 𝑧]𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦)
4		sbv 2091	. . . . 5 ⊢ ([𝑦 / 𝑧]𝜑 ↔ 𝜑)
5	3, 4	imbi12i 350	. . . 4 ⊢ (([𝑦 / 𝑧]𝑥 ∈ 𝑧 → [𝑦 / 𝑧]𝜑) ↔ (𝑥 ∈ 𝑦 → 𝜑))
6	2, 5	bitri 274	. . 3 ⊢ ([𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ (𝑥 ∈ 𝑦 → 𝜑))
7	6	albii 1821	. 2 ⊢ (∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝜑))
8		elequ1 2113	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
9		sbralie.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
10	8, 9	imbi12d 344	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑧 → 𝜑) ↔ (𝑦 ∈ 𝑧 → 𝜓)))
11	10	cbvalvw 2039	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝜓))
12		df-ral 3062	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝜓))
13	12	sbbii 2079	. . . . . 6 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 → 𝜓))
14		sbal 2159	. . . . . 6 ⊢ ([𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 → 𝜓) ↔ ∀𝑦[𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓))
15		sbim 2299	. . . . . . . 8 ⊢ ([𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓) ↔ ([𝑧 / 𝑥]𝑦 ∈ 𝑥 → [𝑧 / 𝑥]𝜓))
16		elsb2 2123	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧)
17		sbv 2091	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝜓 ↔ 𝜓)
18	16, 17	imbi12i 350	. . . . . . . 8 ⊢ (([𝑧 / 𝑥]𝑦 ∈ 𝑥 → [𝑧 / 𝑥]𝜓) ↔ (𝑦 ∈ 𝑧 → 𝜓))
19	15, 18	bitri 274	. . . . . . 7 ⊢ ([𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓) ↔ (𝑦 ∈ 𝑧 → 𝜓))
20	19	albii 1821	. . . . . 6 ⊢ (∀𝑦[𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝜓))
21	13, 14, 20	3bitrri 297	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑧 → 𝜓) ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
22	11, 21	bitri 274	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
23	22	sbbii 2079	. . 3 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑦 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
24		sbal 2159	. . 3 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑))
25		sbco2vv 2100	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
26	23, 24, 25	3bitr3i 300	. 2 ⊢ (∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
27	1, 7, 26	3bitr2i 298	1 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539  [wsb 2067  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-ral 3062

This theorem is referenced by:  tfinds2  7855