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Mirrors > Home > ILE Home > Th. List > ceqsrex2v | GIF version |
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.) |
Ref | Expression |
---|---|
ceqsrex2v.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ceqsrex2v.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ceqsrex2v | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 399 | . . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑))) | |
2 | 1 | rexbii 2445 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑))) |
3 | r19.42v 2591 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑))) | |
4 | 2, 3 | bitri 183 | . . . 4 ⊢ (∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑))) |
5 | 4 | rexbii 2445 | . . 3 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐶 (𝑥 = 𝐴 ∧ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑))) |
6 | ceqsrex2v.1 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 6 | anbi2d 460 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑦 = 𝐵 ∧ 𝜓))) |
8 | 7 | rexbidv 2439 | . . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜓))) |
9 | 8 | ceqsrexv 2819 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (∃𝑥 ∈ 𝐶 (𝑥 = 𝐴 ∧ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜓))) |
10 | 5, 9 | syl5bb 191 | . 2 ⊢ (𝐴 ∈ 𝐶 → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜓))) |
11 | ceqsrex2v.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
12 | 11 | ceqsrexv 2819 | . 2 ⊢ (𝐵 ∈ 𝐷 → (∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜓) ↔ 𝜒)) |
13 | 10, 12 | sylan9bb 458 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ∃wrex 2418 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 |
This theorem is referenced by: (None) |
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