Theorem sbralieOLD 3320

Description: Obsolete version of sbralie 3318 as of 13-Nov-2025. (Contributed by NM, 5-Sep-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
sbralieOLD	⊢ (∀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 sbralieOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 3055	. 2 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝜑))
2		nfv 1921	. . . . 5 ⊢ Ⅎ𝑧𝜑
3	2	sblim 2316	. . . 4 ⊢ ([𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ ([𝑦 / 𝑧]𝑥 ∈ 𝑧 → 𝜑))
4		elsb2 2136	. . . . 5 ⊢ ([𝑦 / 𝑧]𝑥 ∈ 𝑧 ↔ 𝑥 ∈ 𝑦)
5	4	imbi1i 350	. . . 4 ⊢ (([𝑦 / 𝑧]𝑥 ∈ 𝑧 → 𝜑) ↔ (𝑥 ∈ 𝑦 → 𝜑))
6	3, 5	bitri 276	. . 3 ⊢ ([𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ (𝑥 ∈ 𝑦 → 𝜑))
7	6	albii 1826	. 2 ⊢ (∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝑦 → 𝜑))
8		elequ1 2126	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
9		sbralie.1	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
10	8, 9	imbi12d 345	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑧 → 𝜑) ↔ (𝑦 ∈ 𝑧 → 𝜓)))
11	10	cbvalvw 2043	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝜓))
12		df-ral 3055	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝜓))
13	12	sbbii 2087	. . . . . 6 ⊢ ([𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 → 𝜓))
14		sbal 2180	. . . . . 6 ⊢ ([𝑧 / 𝑥]∀𝑦(𝑦 ∈ 𝑥 → 𝜓) ↔ ∀𝑦[𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓))
15		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
16	15	sblim 2316	. . . . . . . 8 ⊢ ([𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓) ↔ ([𝑧 / 𝑥]𝑦 ∈ 𝑥 → 𝜓))
17		elsb2 2136	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧)
18	17	imbi1i 350	. . . . . . . 8 ⊢ (([𝑧 / 𝑥]𝑦 ∈ 𝑥 → 𝜓) ↔ (𝑦 ∈ 𝑧 → 𝜓))
19	16, 18	bitri 276	. . . . . . 7 ⊢ ([𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓) ↔ (𝑦 ∈ 𝑧 → 𝜓))
20	19	albii 1826	. . . . . 6 ⊢ (∀𝑦[𝑧 / 𝑥](𝑦 ∈ 𝑥 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝑧 → 𝜓))
21	13, 14, 20	3bitrri 299	. . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑧 → 𝜓) ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
22	11, 21	bitri 276	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
23	22	sbbii 2087	. . 3 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑦 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
24		sbal 2180	. . 3 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 ∈ 𝑧 → 𝜑) ↔ ∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑))
25		sbco2vv 2110	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
26	23, 24, 25	3bitr3i 302	. 2 ⊢ (∀𝑥[𝑦 / 𝑧](𝑥 ∈ 𝑧 → 𝜑) ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)
27	1, 7, 26	3bitr2i 300	1 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545  [wsb 2073  ∀wral 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-nf 1791  df-sb 2074  df-ral 3055

This theorem is referenced by: (None)