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Mirrors > Home > ILE Home > Th. List > ceqsex2v | GIF version |
Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Ref | Expression |
---|---|
ceqsex2v.1 | ⊢ 𝐴 ∈ V |
ceqsex2v.2 | ⊢ 𝐵 ∈ V |
ceqsex2v.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ceqsex2v.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ceqsex2v | ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1461 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | nfv 1461 | . 2 ⊢ Ⅎ𝑦𝜒 | |
3 | ceqsex2v.1 | . 2 ⊢ 𝐴 ∈ V | |
4 | ceqsex2v.2 | . 2 ⊢ 𝐵 ∈ V | |
5 | ceqsex2v.3 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | ceqsex2v.4 | . 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
7 | 1, 2, 3, 4, 5, 6 | ceqsex2 2639 | 1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∧ w3a 919 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 |
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-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 |
This theorem is referenced by: ceqsex3v 2641 ceqsex4v 2642 |
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