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| Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2110. (Revised by GG, 1-Dec-2023.) | 
| Ref | Expression | 
|---|---|
| ralab2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| rexab2 | ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓)) | |
| 2 | nfsab1 2722 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑} | |
| 3 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
| 4 | 2, 3 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) | 
| 5 | nfv 1914 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜒) | |
| 6 | eleq1ab 2716 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ 𝜑})) | |
| 7 | abid 2718 | . . . . 5 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
| 8 | 6, 7 | bitrdi 287 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)) | 
| 9 | ralab2.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
| 10 | 8, 9 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) | 
| 11 | 4, 5, 10 | cbvexv1 2344 | . 2 ⊢ (∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑 ∧ 𝜒)) | 
| 12 | 1, 11 | bitri 275 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∃wrex 3070 | 
| 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-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-rex 3071 | 
| This theorem is referenced by: rexrab2 3706 tmdgsum2 24104 clrellem 43635 brtrclfv2 43740 | 
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