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Mirrors > Home > MPE Home > Th. List > disjeq1d | Structured version Visualization version GIF version |
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
disjeq1d | ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | disjeq1 5029 | . 2 ⊢ (𝐴 = 𝐵 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 Disj wdisj 5022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-clab 2797 df-cleq 2811 df-clel 2890 df-rmo 3143 df-in 3940 df-ss 3949 df-disj 5023 |
This theorem is referenced by: disjeq12d 5031 disjxiun 5054 disjdifprg 30253 disjdifprg2 30254 disjun0 30273 tocyccntz 30713 measxun2 31368 measssd 31373 meadjun 42621 |
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