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Mirrors > Home > ILE Home > Th. List > disjeq12d | GIF version |
Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
disjeq12d.1 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
disjeq12d | ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | disjeq1d 3922 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
3 | disjeq12d.1 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
5 | 4 | disjeq2dv 3919 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
6 | 2, 5 | bitrd 187 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 Disj wdisj 3914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-ral 2422 df-rmo 2425 df-in 3082 df-ss 3089 df-disj 3915 |
This theorem is referenced by: (None) |
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