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Mirrors > Home > ILE Home > Th. List > rsp2e | GIF version |
Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.) |
Ref | Expression |
---|---|
rsp2e | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 997 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 ∈ 𝐴) | |
2 | rspe 2526 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑) | |
3 | 2 | 3adant1 1015 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑) |
4 | 19.8a 1590 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑)) | |
5 | 1, 3, 4 | syl2anc 411 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑)) |
6 | df-rex 2461 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑)) | |
7 | 5, 6 | sylibr 134 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 |
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-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-rex 2461 |
This theorem is referenced by: (None) |
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