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Mirrors > Home > MPE Home > Th. List > unieqOLD | Structured version Visualization version GIF version |
Description: Obsolete version of unieq 4842 as of 13-Aprl-2024. (Contributed by NM, 10-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) 29-Jun-2011.) |
Ref | Expression |
---|---|
unieqOLD | ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 3405 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)) | |
2 | 1 | abbidv 2884 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}) |
3 | dfuni2 4833 | . 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} | |
4 | dfuni2 4833 | . 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥} | |
5 | 2, 3, 4 | 3eqtr4g 2880 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 {cab 2798 ∃wrex 3138 ∪ cuni 4831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-rex 3143 df-uni 4832 |
This theorem is referenced by: (None) |
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