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Mirrors > Home > MPE Home > Th. List > unab | Structured version Visualization version GIF version |
Description: Union of two class abstractions. (Contributed by NM, 29-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
unab | ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∨ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbor 2312 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) | |
2 | df-clab 2777 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∨ 𝜓)} ↔ [𝑦 / 𝑥](𝜑 ∨ 𝜓)) | |
3 | df-clab 2777 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
4 | df-clab 2777 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
5 | 3, 4 | orbi12i 912 | . . 3 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓)) |
6 | 1, 2, 5 | 3bitr4ri 307 | . 2 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ 𝑦 ∈ {𝑥 ∣ (𝜑 ∨ 𝜓)}) |
7 | 6 | uneqri 4078 | 1 ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑥 ∣ (𝜑 ∨ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1538 [wsb 2069 ∈ wcel 2111 {cab 2776 ∪ cun 3879 |
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-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-un 3886 |
This theorem is referenced by: unrab 4226 rabun2 4234 dfif6 4428 unopab 5109 dmun 5743 hashf1lem2 13810 vdwlem6 16312 vtxdun 27271 satfvsuclem1 32719 satf0suclem 32735 fmlasuc0 32744 diophun 39714 |
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