Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > disjeq12d | 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) |
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 5039 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)) |
3 | disjeq12d.1 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
5 | 4 | disjeq2dv 5036 | . 2 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
6 | 2, 5 | bitrd 281 | 1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Disj wdisj 5031 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-rmo 3146 df-in 3943 df-ss 3952 df-disj 5032 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |