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Mirrors > Home > ILE Home > Th. List > rexcom4b | GIF version |
Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Ref | Expression |
---|---|
rexcom4b.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
rexcom4b | ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4a 2578 | . 2 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) | |
2 | rexcom4b.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
3 | 2 | isseti 2563 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝐵 |
4 | 3 | biantru 286 | . . 3 ⊢ (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
5 | 4 | rexbii 2331 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵)) |
6 | 1, 5 | bitr4i 176 | 1 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵) ↔ ∃𝑦 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃wrex 2307 Vcvv 2557 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 |
This theorem is referenced by: (None) |
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