Theorem cbvralf 2721

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

Ref	Expression
cbvralf.1	⊢ Ⅎ𝑥𝐴
cbvralf.2	⊢ Ⅎ𝑦𝐴
cbvralf.3	⊢ Ⅎ𝑦𝜑
cbvralf.4	⊢ Ⅎ𝑥𝜓
cbvralf.5	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvralf	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Proof of Theorem cbvralf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1542	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝜑)
2		cbvralf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2333	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4		nfs1v 1958	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
5	3, 4	nfim 1586	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)
6		eleq1 2259	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
7		sbequ12 1785	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	6, 7	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)))
9	1, 5, 8	cbval 1768	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑))
10		cbvralf.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2333	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
12		cbvralf.3	. . . . . 6 ⊢ Ⅎ𝑦𝜑
13	12	nfsb 1965	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
14	11, 13	nfim 1586	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)
15		nfv 1542	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 → 𝜓)
16		eleq1 2259	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
17		sbequ 1854	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18		cbvralf.4	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
19		cbvralf.5	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
20	18, 19	sbie 1805	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
21	17, 20	bitrdi 196	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
22	16, 21	imbi12d 234	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
23	14, 15, 22	cbval 1768	. . 3 ⊢ (∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
24	9, 23	bitri 184	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
25		df-ral 2480	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
26		df-ral 2480	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
27	24, 25, 26	3bitr4i 212	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  Ⅎwnf 1474  [wsb 1776   ∈ wcel 2167  Ⅎwnfc 2326  ∀wral 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480

This theorem is referenced by:  cbvral  2725  ffnfvf  5721