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Mirrors > Home > ILE Home > Th. List > rexab | GIF version |
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
ralab.1 | ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexab | ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2461 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒)) | |
2 | vex 2740 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | ralab.1 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 2881 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓) |
5 | 4 | anbi1i 458 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)) |
6 | 5 | exbii 1605 | . 2 ⊢ (∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒) ↔ ∃𝑥(𝜓 ∧ 𝜒)) |
7 | 1, 6 | bitri 184 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∃wex 1492 ∈ wcel 2148 {cab 2163 ∃wrex 2456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 |
This theorem is referenced by: rexrnmpo 5983 |
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