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Mirrors > Home > ILE Home > Th. List > ceqsrexbv | GIF version |
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) |
Ref | Expression |
---|---|
ceqsrexv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
ceqsrexbv | ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.42v 2565 | . 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝐴 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑))) | |
2 | eleq1 2180 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 2 | adantr 274 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
4 | 3 | pm5.32ri 450 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑))) |
5 | 4 | bicomi 131 | . . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑))) |
6 | 5 | baib 889 | . . 3 ⊢ (𝑥 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ 𝜑))) |
7 | 6 | rexbiia 2427 | . 2 ⊢ (∃𝑥 ∈ 𝐵 (𝐴 ∈ 𝐵 ∧ (𝑥 = 𝐴 ∧ 𝜑)) ↔ ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑)) |
8 | ceqsrexv.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
9 | 8 | ceqsrexv 2789 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
10 | 9 | pm5.32i 449 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑)) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
11 | 1, 7, 10 | 3bitr3i 209 | 1 ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∃wrex 2394 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 |
This theorem is referenced by: frecsuclem 6271 |
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