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| 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 2896 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 | 
| 3 | cbvdisj.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
| 4 | 3 | nfcri 2896 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 | 
| 5 | cbvdisj.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
| 6 | 5 | eleq2d 2826 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) | 
| 7 | 2, 4, 6 | cbvrmow 3408 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | 
| 8 | 7 | albii 1818 | . 2 ⊢ (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | 
| 9 | df-disj 5110 | . 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
| 10 | df-disj 5110 | . 2 ⊢ (Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀𝑧∃*𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) | |
| 11 | 8, 9, 10 | 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 Ⅎwnfc 2889 ∃*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-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-mo 2539 df-cleq 2728 df-clel 2815 df-nfc 2891 df-rmo 3379 df-disj 5110 | 
| This theorem is referenced by: disjors 5125 disjxiun 5139 volfiniun 25583 voliun 25590 carsggect 34321 omsmeas 34326 disjf1 45193 disjrnmpt2 45198 fsumiunss 45595 sge0iunmpt 46438 iundjiun 46480 meadjiun 46486 | 
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