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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 1830 | . . . . . 6 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
2 | cbvdisjf.2 | . . . . . . 7 ⊢ Ⅎ𝑦𝐵 | |
3 | 2 | nfcri 2745 | . . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
4 | 1, 3 | nfan 1816 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) |
5 | cbvdisjf.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
6 | 5 | nfcri 2745 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
7 | cbvdisjf.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝐶 | |
8 | 7 | nfcri 2745 | . . . . . 6 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
9 | 6, 8 | nfan 1816 | . . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) |
10 | eleq1 2676 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
11 | cbvdisjf.4 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
12 | 11 | eleq2d 2673 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
13 | 10, 12 | anbi12d 743 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶))) |
14 | 4, 9, 13 | cbvmo 2494 | . . . 4 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) |
15 | df-rmo 2904 | . . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) | |
16 | df-rmo 2904 | . . . 4 ⊢ (∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶)) | |
17 | 14, 15, 16 | 3bitr4i 291 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
18 | 17 | albii 1737 | . 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
19 | df-disj 4554 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
20 | df-disj 4554 | . 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | |
21 | 18, 19, 20 | 3bitr4i 291 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 Ⅎwnfc 2738 ∃*wrmo 2899 Disj wdisj 4553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rmo 2904 df-disj 4554 |
This theorem is referenced by: disjorsf 28775 ldgenpisyslem1 29553 |
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