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Mirrors > Home > MPE Home > Th. List > unabw | Structured version Visualization version GIF version |
Description: Union of two class abstractions. Version of unab 4314 using implicit substitution, which does not require ax-8 2108, ax-10 2139, ax-12 2175. (Contributed by GG, 15-Oct-2024.) |
Ref | Expression |
---|---|
unabw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
unabw.2 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
unabw | ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝜒 ∨ 𝜃)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-un 3968 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓})} | |
2 | df-clab 2713 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
3 | unabw.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | 3 | sbievw 2091 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒) |
5 | 2, 4 | bitri 275 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒) |
6 | df-clab 2713 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
7 | unabw.2 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
8 | 7 | sbievw 2091 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜃) |
9 | 6, 8 | bitri 275 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝜃) |
10 | 5, 9 | orbi12i 914 | . . 3 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜒 ∨ 𝜃)) |
11 | 10 | abbii 2807 | . 2 ⊢ {𝑦 ∣ (𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓})} = {𝑦 ∣ (𝜒 ∨ 𝜃)} |
12 | 1, 11 | eqtri 2763 | 1 ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝜒 ∨ 𝜃)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1537 [wsb 2062 ∈ wcel 2106 {cab 2712 ∪ cun 3961 |
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 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-un 3968 |
This theorem is referenced by: dfif6 4534 unopab 5230 |
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