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| Description: Union of two class abstractions. Version of unab 4308 using implicit substitution, which does not require ax-8 2110, ax-10 2141, ax-12 2177. (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 3956 | . 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 2093 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜒) | 
| 5 | 2, 4 | bitri 275 | . . . 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 2093 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜃) | 
| 9 | 6, 8 | bitri 275 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ 𝜃) | 
| 10 | 5, 9 | orbi12i 915 | . . 3 ⊢ ((𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜒 ∨ 𝜃)) | 
| 11 | 10 | abbii 2809 | . 2 ⊢ {𝑦 ∣ (𝑦 ∈ {𝑥 ∣ 𝜑} ∨ 𝑦 ∈ {𝑥 ∣ 𝜓})} = {𝑦 ∣ (𝜒 ∨ 𝜃)} | 
| 12 | 1, 11 | eqtri 2765 | 1 ⊢ ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝜒 ∨ 𝜃)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∨ wo 848 = wceq 1540 [wsb 2064 ∈ wcel 2108 {cab 2714 ∪ cun 3949 | 
| 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 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-un 3956 | 
| This theorem is referenced by: dfif6 4528 unopab 5224 | 
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