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Mirrors > Home > ILE Home > Th. List > abexex | GIF version |
Description: A condition where a class builder 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 2441 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | abexex.2 | . . . . . 6 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
3 | 2 | pm4.71ri 390 | . . . . 5 ⊢ (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
4 | 3 | exbii 1585 | . . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 1, 4 | bitr4i 186 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥𝜑) |
6 | 5 | abbii 2273 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑} |
7 | abexex.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | abexex.3 | . . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V | |
9 | 7, 8 | abrexex2 6073 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V |
10 | 6, 9 | eqeltrri 2231 | 1 ⊢ {𝑦 ∣ ∃𝑥𝜑} ∈ V |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∃wex 1472 ∈ wcel 2128 {cab 2143 ∃wrex 2436 Vcvv 2712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 |
This theorem is referenced by: (None) |
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