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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 3068 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | abexex.2 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
3 | 2 | pm4.71ri 564 | . . . . 5 ⊢ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
4 | 3 | exbii 1855 | . . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 1, 4 | bitr4i 281 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥𝜑) |
6 | 5 | abbii 2809 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑} |
7 | abexex.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | abexex.3 | . . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V | |
9 | 7, 8 | abrexex2 7760 | . 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 399 ∃wex 1787 ∈ wcel 2111 {cab 2715 ∃wrex 3063 Vcvv 3420 |
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 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pr 5336 ax-un 7541 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 |
This theorem is referenced by: brdom7disj 10169 brdom6disj 10170 |
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