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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 5122 | . 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 206 = wceq 1537 Disj wdisj 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-mo 2538 df-cleq 2727 df-clel 2814 df-rmo 3378 df-ss 3980 df-disj 5116 |
This theorem is referenced by: disjeq12d 5124 disjxiun 5145 disjdifprg 32595 disjdifprg2 32596 disjun0 32615 tocyccntz 33147 measxun2 34191 measssd 34196 meadjun 46418 |
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