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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 4908 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
3 | iuneq12d.2 | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷) |
5 | 4 | iuneq2dv 4905 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
6 | 2, 5 | eqtrd 2833 | 1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∪ ciun 4881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-in 3888 df-ss 3898 df-iun 4883 |
This theorem is referenced by: disjiunb 5019 cfsmolem 9681 cfsmo 9682 wunex2 10149 wuncval2 10158 imasval 16776 lpival 20011 cnextval 22666 cnextfval 22667 dvfval 24500 fedgmullem1 31113 mblfinlem2 35095 heiborlem10 35258 iunrelexpmin1 40409 iunrelexpmin2 40413 colleq12d 40961 |
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