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Mirrors > Home > ILE Home > Th. List > rexpr | GIF version |
Description: Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralpr.1 | ⊢ 𝐴 ∈ V |
ralpr.2 | ⊢ 𝐵 ∈ V |
ralpr.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ralpr.4 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexpr | ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralpr.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | ralpr.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | ralpr.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | ralpr.4 | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
5 | 3, 4 | rexprg 3444 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
6 | 1, 2, 5 | mp2an 416 | 1 ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∨ wo 661 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 Vcvv 2601 {cpr 3399 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-sn 3404 df-pr 3405 |
This theorem is referenced by: (None) |
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