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Mirrors > Home > ILE Home > Th. List > spc2ev | GIF version |
Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.) |
Ref | Expression |
---|---|
spc2ev.1 | ⊢ 𝐴 ∈ V |
spc2ev.2 | ⊢ 𝐵 ∈ V |
spc2ev.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spc2ev | ⊢ (𝜓 → ∃𝑥∃𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spc2ev.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | spc2ev.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | spc2ev.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
4 | 3 | spc2egv 2779 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥∃𝑦𝜑)) |
5 | 1, 2, 4 | mp2an 423 | 1 ⊢ (𝜓 → ∃𝑥∃𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-v 2691 |
This theorem is referenced by: relop 4697 th3qlem2 6540 endisj 6726 axcnre 7713 |
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