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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 2213 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
3 | cbvdisj.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfcri 2213 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
5 | cbvdisj.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
6 | 5 | eleq2d 2148 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
7 | 2, 4, 6 | cbvrmo 2576 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
8 | 7 | albii 1399 | . 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
9 | df-disj 3767 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
10 | df-disj 3767 | . 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | |
11 | 8, 9, 10 | 3bitr4i 210 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 = wceq 1284 ∈ wcel 1433 Ⅎwnfc 2206 ∃*wrmo 2351 Disj wdisj 3766 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-reu 2355 df-rmo 2356 df-disj 3767 |
This theorem is referenced by: cbvdisjv 3777 |
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