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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 3161 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | 1 | anim1d 336 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑))) |
3 | 2 | reximdv2 2586 | 1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 ∃wrex 2466 ⊆ wss 3141 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-11 1516 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-rex 2471 df-in 3147 df-ss 3154 |
This theorem is referenced by: iunss1 3909 moriotass 5872 tfr1onlemssrecs 6354 tfrcllemssrecs 6367 fiss 6990 supelti 7015 ctssdclemn0 7123 ctssdc 7126 enumctlemm 7127 nninfwlpoimlemginf 7188 rerecapb 8814 lbzbi 9630 fiubm 10822 rexico 11244 alzdvds 11874 zsupcl 11962 infssuzex 11964 gcddvds 11978 dvdslegcd 11979 pclemub 12301 subrgdvds 13455 ssrest 13978 reeff1olem 14488 bj-charfunbi 14859 bj-nn0suc 15012 |
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