Theorem cbvrald 10598

Description: Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)

Hypotheses

Ref	Expression
cbvrald.nf0	⊢ Ⅎ𝑥𝜑
cbvrald.nf1	⊢ Ⅎ𝑦𝜑
cbvrald.nf2	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbvrald.nf3	⊢ (𝜑 → Ⅎ𝑥𝜒)
cbvrald.is	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbvrald	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

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

Proof of Theorem cbvrald

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvrald.nf0	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nfv 1461	. . . 4 ⊢ Ⅎ𝑧𝜑
3		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
4	3	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑧 𝑥 ∈ 𝐴)
5		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑧𝜓
6	5	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑧𝜓)
7	4, 6	nfimd 1517	. . . 4 ⊢ (𝜑 → Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝜓))
8		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
9	8	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑧 ∈ 𝐴)
10		nfs1v 1856	. . . . . 6 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜓
11	10	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑥]𝜓)
12	9, 11	nfimd 1517	. . . 4 ⊢ (𝜑 → Ⅎ𝑥(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜓))
13		eleq1 2141	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
14	13	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
15		sbequ12 1694	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
16	15	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
17	14, 16	imbi12d 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑧) → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜓)))
18	17	ex 113	. . . 4 ⊢ (𝜑 → (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜓))))
19	1, 2, 7, 12, 18	cbv2 1675	. . 3 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜓)))
20		cbvrald.nf1	. . . 4 ⊢ Ⅎ𝑦𝜑
21		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
22	21	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦 𝑧 ∈ 𝐴)
23		cbvrald.nf2	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑦𝜓)
24	1, 23	nfsbd 1892	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦[𝑧 / 𝑥]𝜓)
25	22, 24	nfimd 1517	. . . 4 ⊢ (𝜑 → Ⅎ𝑦(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜓))
26		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐴
27	26	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑧 𝑦 ∈ 𝐴)
28		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑧𝜒
29	28	a1i 9	. . . . 5 ⊢ (𝜑 → Ⅎ𝑧𝜒)
30	27, 29	nfimd 1517	. . . 4 ⊢ (𝜑 → Ⅎ𝑧(𝑦 ∈ 𝐴 → 𝜒))
31		eleq1 2141	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
32	31	adantl 271	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑦) → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
33		sbequ 1761	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))
34		cbvrald.nf3	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝜒)
35		cbvrald.is	. . . . . . . 8 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
36	1, 34, 35	sbied 1711	. . . . . . 7 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
37	33, 36	sylan9bbr 450	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑦) → ([𝑧 / 𝑥]𝜓 ↔ 𝜒))
38	32, 37	imbi12d 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑦) → ((𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜓) ↔ (𝑦 ∈ 𝐴 → 𝜒)))
39	38	ex 113	. . . 4 ⊢ (𝜑 → (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜓) ↔ (𝑦 ∈ 𝐴 → 𝜒))))
40	2, 20, 25, 30, 39	cbv2 1675	. . 3 ⊢ (𝜑 → (∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜒)))
41	19, 40	bitrd 186	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜒)))
42		df-ral 2353	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
43		df-ral 2353	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜒 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜒))
44	41, 42, 43	3bitr4g 221	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282  Ⅎwnf 1389   ∈ wcel 1433  [wsb 1685  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-ral 2353

This theorem is referenced by:  setindft  10760