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Mirrors > Home > MPE Home > Th. List > cbvdisjv | Structured version Visualization version GIF version |
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 | nfcv 2764 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvdisjv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvdisj 4630 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 Disj wdisj 4620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-disj 4621 |
This theorem is referenced by: uniioombllem4 23354 hashunif 29562 totprob 30489 disjrnmpt2 39375 ismeannd 40684 psmeasure 40688 volmea 40691 meaiuninclem 40697 caratheodorylem1 40740 caratheodory 40742 |
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