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Mirrors > Home > MPE Home > Th. List > rexab | Structured version Visualization version 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 3141 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒)) | |
2 | vex 3495 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | ralab.1 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | elab 3664 | . . . 4 ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓) |
5 | 4 | anbi1i 623 | . . 3 ⊢ ((𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒) ↔ (𝜓 ∧ 𝜒)) |
6 | 5 | exbii 1839 | . 2 ⊢ (∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜒) ↔ ∃𝑥(𝜓 ∧ 𝜒)) |
7 | 1, 6 | bitri 276 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜒 ↔ ∃𝑥(𝜓 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 {cab 2796 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 |
This theorem is referenced by: 4sqlem12 16280 mblfinlem3 34812 mblfinlem4 34813 ismblfin 34814 itg2addnclem 34824 itg2addnc 34827 diophrex 39250 |
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