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Mirrors > Home > ILE Home > Th. List > rexab2 | 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 |
---|---|
rexab2 | ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2441 | . 2 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓)) | |
2 | nfsab1 2147 | . . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ {𝑦 ∣ 𝜑} | |
3 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑦𝜓 | |
4 | 2, 3 | nfan 1545 | . . 3 ⊢ Ⅎ𝑦(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) |
5 | nfv 1508 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜒) | |
6 | eleq1 2220 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ 𝜑})) | |
7 | abid 2145 | . . . . 5 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
8 | 6, 7 | bitrdi 195 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)) |
9 | ralab2.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
10 | 8, 9 | anbi12d 465 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))) |
11 | 4, 5, 10 | cbvex 1736 | . 2 ⊢ (∃𝑥(𝑥 ∈ {𝑦 ∣ 𝜑} ∧ 𝜓) ↔ ∃𝑦(𝜑 ∧ 𝜒)) |
12 | 1, 11 | bitri 183 | 1 ⊢ (∃𝑥 ∈ {𝑦 ∣ 𝜑}𝜓 ↔ ∃𝑦(𝜑 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∃wex 1472 ∈ wcel 2128 {cab 2143 ∃wrex 2436 |
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-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-rex 2441 |
This theorem is referenced by: rexrab2 2879 |
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