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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 5115 | . 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 205 = wceq 1533 Disj wdisj 5108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-mo 2528 df-cleq 2717 df-clel 2802 df-rmo 3364 df-ss 3956 df-disj 5109 |
This theorem is referenced by: disjeq12d 5117 disjxiun 5140 disjdifprg 32410 disjdifprg2 32411 disjun0 32430 tocyccntz 32910 measxun2 33886 measssd 33891 meadjun 45913 |
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