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| Mirrors > Home > MPE Home > Th. List > abexex | Structured version Visualization version GIF version | ||
| Description: A condition where a class abstraction continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.) | 
| Ref | Expression | 
|---|---|
| abexex.1 | ⊢ 𝐴 ∈ V | 
| abexex.2 | ⊢ (𝜑 → 𝑥 ∈ 𝐴) | 
| abexex.3 | ⊢ {𝑦 ∣ 𝜑} ∈ V | 
| Ref | Expression | 
|---|---|
| abexex | ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-rex 3070 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | abexex.2 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
| 3 | 2 | pm4.71ri 560 | . . . . 5 ⊢ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) | 
| 4 | 3 | exbii 1847 | . . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | 
| 5 | 1, 4 | bitr4i 278 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥𝜑) | 
| 6 | 5 | abbii 2808 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑} | 
| 7 | abexex.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 8 | abexex.3 | . . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V | |
| 9 | 7, 8 | abrexex2 7995 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V | 
| 10 | 6, 9 | eqeltrri 2837 | 1 ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∃wex 1778 ∈ wcel 2107 {cab 2713 ∃wrex 3069 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-v 3481 df-ss 3967 df-uni 4907 df-iun 4992 | 
| This theorem is referenced by: brdom7disj 10572 brdom6disj 10573 | 
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