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| Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016.) | 
| Ref | Expression | 
|---|---|
| cbvdisjv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| cbvdisjv | ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cbvdisjv.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 2 | 1 | eleq2d 2826 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) | 
| 3 | 2 | cbvrmovw 3402 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | 
| 4 | 3 | albii 1818 | . 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | 
| 5 | df-disj 5110 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
| 6 | df-disj 5110 | . 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | |
| 7 | 4, 5, 6 | 3bitr4i 303 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 = wceq 1539 ∈ wcel 2107 ∃*wrmo 3378 Disj wdisj 5109 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-mo 2539 df-cleq 2728 df-clel 2815 df-rmo 3379 df-disj 5110 | 
| This theorem is referenced by: uniioombllem4 25622 hashunif 32811 tocyccntz 33165 totprob 34430 disjrnmpt2 45198 ismeannd 46487 psmeasure 46491 volmea 46494 meaiuninclem 46500 caratheodorylem1 46546 caratheodory 46548 | 
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