Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvdisjf | Structured version Visualization version GIF version |
Description: Change bound variables in a disjoint collection. (Contributed by Thierry Arnoux, 6-Apr-2017.) |
Ref | Expression |
---|---|
cbvdisjf.1 | ⊢ Ⅎ𝑥𝐴 |
cbvdisjf.2 | ⊢ Ⅎ𝑦𝐵 |
cbvdisjf.3 | ⊢ Ⅎ𝑥𝐶 |
cbvdisjf.4 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvdisjf | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
2 | cbvdisjf.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐵 | |
3 | 2 | nfcri 2971 | . . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
4 | 1, 3 | nfan 1896 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
5 | cbvdisjf.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
6 | 5 | nfcri 2971 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
7 | cbvdisjf.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝐶 | |
8 | 7 | nfcri 2971 | . . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
9 | 6, 8 | nfan 1896 | . . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) |
10 | eleq1w 2895 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
11 | cbvdisjf.4 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
12 | 11 | eleq2d 2898 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
13 | 10, 12 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))) |
14 | 4, 9, 13 | cbvmow 2684 | . . . 4 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) |
15 | df-rmo 3146 | . . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
16 | df-rmo 3146 | . . . 4 ⊢ (∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) | |
17 | 14, 15, 16 | 3bitr4i 305 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
18 | 17 | albii 1816 | . 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
19 | df-disj 5025 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
20 | df-disj 5025 | . 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | |
21 | 18, 19, 20 | 3bitr4i 305 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∃*wmo 2616 Ⅎwnfc 2961 ∃*wrmo 3141 Disj wdisj 5024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rmo 3146 df-disj 5025 |
This theorem is referenced by: disjorsf 30324 ldgenpisyslem1 31417 |
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