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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 1491 | . 2 ⊢ Ⅎ𝑥𝜓 | |
2 | nfv 1491 | . 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 2697 | 1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 945 = wceq 1314 ∃wex 1451 ∈ wcel 1463 Vcvv 2657 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-v 2659 |
This theorem is referenced by: ceqsex3v 2699 ceqsex4v 2700 |
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