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Mirrors > Home > MPE Home > Th. List > iuneq12d | Structured version Visualization version GIF version |
Description: Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.) |
Ref | Expression |
---|---|
iuneq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
iuneq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
iuneq12d | ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | iuneq1d 5028 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
3 | iuneq12d.2 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
5 | 4 | iuneq2dv 5025 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
6 | 2, 5 | eqtrd 2766 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∪ ciun 5001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-v 3464 df-ss 3964 df-iun 5003 |
This theorem is referenced by: disjiunb 5142 cfsmolem 10313 cfsmo 10314 wunex2 10781 wuncval2 10790 imasval 17526 lpival 21313 cnextval 24056 cnextfval 24057 dvfval 25917 fedgmullem1 33524 irngval 33561 mblfinlem2 37359 heiborlem10 37521 iunrelexpmin1 43375 iunrelexpmin2 43379 colleq12d 43927 |
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