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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 4199 using implicit substitution, which does not require ax-8 2114, ax-10 2143, ax-12 2177. (Contributed by Gino Giotto, 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 3858 | . 2 ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓})} | |
2 | df-clab 2715 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
3 | unabw.1 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | 3 | sbievw 2101 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒) |
5 | 2, 4 | bitri 278 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒) |
6 | df-clab 2715 | . . . . 5 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
7 | unabw.2 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃)) | |
8 | 7 | sbievw 2101 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜃) |
9 | 6, 8 | bitri 278 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝜃) |
10 | 5, 9 | orbi12i 915 | . . 3 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜒 ∨ 𝜃)) |
11 | 10 | abbii 2801 | . 2 ⊢ {𝑦 ∣ (𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓})} = {𝑦 ∣ (𝜒 ∨ 𝜃)} |
12 | 1, 11 | eqtri 2759 | 1 ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝜒 ∨ 𝜃)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∨ wo 847 = wceq 1543 [wsb 2072 ∈ wcel 2112 {cab 2714 ∪ cun 3851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-un 3858 |
This theorem is referenced by: dfif6 4428 unopab 5119 |
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