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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 3069 | . . . 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 1845 | . . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 1, 4 | bitr4i 278 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥𝜑) |
6 | 5 | abbii 2807 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑} |
7 | abexex.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | abexex.3 | . . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V | |
9 | 7, 8 | abrexex2 7993 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V |
10 | 6, 9 | eqeltrri 2836 | 1 ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 {cab 2712 ∃wrex 3068 Vcvv 3478 |
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-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-v 3480 df-ss 3980 df-uni 4913 df-iun 4998 |
This theorem is referenced by: brdom7disj 10569 brdom6disj 10570 |
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