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Mirrors > Home > ILE Home > Th. List > cbvdisj | GIF version |
Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
cbvdisj.1 | ⊢ Ⅎ𝑦𝐵 |
cbvdisj.2 | ⊢ Ⅎ𝑥𝐶 |
cbvdisj.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvdisj | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvdisj.1 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
2 | 1 | nfcri 2313 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
3 | cbvdisj.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfcri 2313 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
5 | cbvdisj.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
6 | 5 | eleq2d 2247 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
7 | 2, 4, 6 | cbvrmo 2702 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
8 | 7 | albii 1470 | . 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
9 | df-disj 3978 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
10 | df-disj 3978 | . 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | |
11 | 8, 9, 10 | 3bitr4i 212 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 = wceq 1353 ∈ wcel 2148 Ⅎwnfc 2306 ∃*wrmo 2458 Disj wdisj 3977 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-reu 2462 df-rmo 2463 df-disj 3978 |
This theorem is referenced by: cbvdisjv 3988 disjnims 3992 |
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