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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 3985 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
3 | disjeq12d.1 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | adantr 276 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
5 | 4 | disjeq2dv 3982 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
6 | 2, 5 | bitrd 188 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Disj wdisj 3977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-ral 2460 df-rmo 2463 df-in 3135 df-ss 3142 df-disj 3978 |
This theorem is referenced by: (None) |
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