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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 2977 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2977 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvdisjv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvdisj 5041 | 1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 Disj wdisj 5031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-disj 5032 |
This theorem is referenced by: uniioombllem4 24187 hashunif 30528 tocyccntz 30786 totprob 31685 disjrnmpt2 41469 ismeannd 42769 psmeasure 42773 volmea 42776 meaiuninclem 42782 caratheodorylem1 42828 caratheodory 42830 |
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