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Mirrors > Home > MPE Home > Th. List > cbvralf | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) |
Ref | Expression |
---|---|
cbvralf.1 | ⊢ Ⅎ𝑥𝐴 |
cbvralf.2 | ⊢ Ⅎ𝑦𝐴 |
cbvralf.3 | ⊢ Ⅎ𝑦𝜑 |
cbvralf.4 | ⊢ Ⅎ𝑥𝜓 |
cbvralf.5 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvralf | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1883 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝜑) | |
2 | cbvralf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2787 | . . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
4 | nfs1v 2465 | . . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
5 | 3, 4 | nfim 1865 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) |
6 | eleq1 2718 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
7 | sbequ12 2149 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
8 | 6, 7 | imbi12d 333 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑))) |
9 | 1, 5, 8 | cbval 2307 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑)) |
10 | cbvralf.2 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
11 | 10 | nfcri 2787 | . . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴 |
12 | cbvralf.3 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
13 | 12 | nfsb 2468 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
14 | 11, 13 | nfim 1865 | . . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) |
15 | nfv 1883 | . . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 → 𝜓) | |
16 | eleq1 2718 | . . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
17 | sbequ 2404 | . . . . . 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 2436 | . . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
21 | 17, 20 | syl6bb 276 | . . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓)) |
22 | 16, 21 | imbi12d 333 | . . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓))) |
23 | 14, 15, 22 | cbval 2307 | . . 3 ⊢ (∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
24 | 9, 23 | bitri 264 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
25 | df-ral 2946 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
26 | df-ral 2946 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
27 | 24, 25, 26 | 3bitr4i 292 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1521 Ⅎwnf 1748 [wsb 1937 ∈ wcel 2030 Ⅎwnfc 2780 ∀wral 2941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 |
This theorem is referenced by: cbvrexf 3196 cbvral 3197 reusv2lem4 4902 reusv2 4904 ffnfvf 6429 nnwof 11792 nnindf 29693 scottexf 34106 scott0f 34107 evth2f 39488 evthf 39500 supxrleubrnmptf 39993 stoweidlem14 40549 stoweidlem28 40563 stoweidlem59 40594 |
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