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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 4939 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
3 | iuneq12d.2 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
5 | 4 | iuneq2dv 4936 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
6 | 2, 5 | eqtrd 2856 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ ciun 4912 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3497 df-in 3943 df-ss 3952 df-iun 4914 |
This theorem is referenced by: disjiunb 5048 cfsmolem 9686 cfsmo 9687 wunex2 10154 wuncval2 10163 imasval 16778 lpival 20012 cnextval 22663 cnextfval 22664 dvfval 24489 fedgmullem1 31020 mblfinlem2 34924 heiborlem10 35092 iunrelexpmin1 40046 iunrelexpmin2 40050 colleq12d 40582 |
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