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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 1539 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | nfv 1539 | . 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 2792 | 1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-v 2754 |
This theorem is referenced by: ceqsex3v 2794 ceqsex4v 2795 |
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