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Mirrors > Home > ILE Home > Th. List > ssrexv | GIF version |
Description: Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.) |
Ref | Expression |
---|---|
ssrexv | ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3041 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | anim1d 332 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
3 | 2 | reximdv2 2490 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1448 ∃wrex 2376 ⊆ wss 3021 |
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 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-rex 2381 df-in 3027 df-ss 3034 |
This theorem is referenced by: iunss1 3771 moriotass 5690 tfr1onlemssrecs 6166 tfrcllemssrecs 6179 supelti 6804 ctssdclemn0 6910 ctssdc 6912 enumctlemm 6913 lbzbi 9258 rexico 10833 alzdvds 11347 zsupcl 11435 infssuzex 11437 gcddvds 11447 dvdslegcd 11448 ssrest 12133 bj-nn0suc 12747 |
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