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Mirrors > Home > MPE Home > Th. List > rexrab2 | Structured version Visualization version GIF version |
Description: Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
ralab2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexrab2 | ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3074 | . . 3 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | rexeqi 3345 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓) |
3 | ralab2.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
4 | 3 | rexab2 3638 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}𝜓 ↔ ∃𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒)) |
5 | anass 468 | . . . 4 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ (𝑦 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒))) | |
6 | 5 | exbii 1853 | . . 3 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒))) |
7 | df-rex 3071 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝜑 ∧ 𝜒))) | |
8 | 6, 7 | bitr4i 277 | . 2 ⊢ (∃𝑦((𝑦 ∈ 𝐴 ∧ 𝜑) ∧ 𝜒) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒)) |
9 | 2, 4, 8 | 3bitri 296 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑}𝜓 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1785 ∈ wcel 2109 {cab 2716 ∃wrex 3066 {crab 3069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-tru 1544 df-ex 1786 df-nf 1790 df-sb 2071 df-clab 2717 df-cleq 2731 df-ral 3070 df-rex 3071 df-rab 3074 |
This theorem is referenced by: frminex 5568 sstotbnd3 35913 |
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