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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 2850 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥∃𝑦𝜑)) |
5 | 1, 2, 4 | mp2an 426 | 1 ⊢ (𝜓 → ∃𝑥∃𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2164 Vcvv 2760 |
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 2175 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-v 2762 |
This theorem is referenced by: relop 4806 th3qlem2 6683 endisj 6869 axcnre 7931 |
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