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Mirrors > Home > MPE Home > Th. List > rspcedv | Structured version Visualization version GIF version |
Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rspcdv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcdv.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rspcedv | ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcdv.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
2 | rspcdv.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | biimprd 237 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) |
4 | 1, 3 | rspcimedv 3284 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 |
This theorem is referenced by: rspcedvd 3289 fsuppmapnn0fiubOLD 12653 wrdl1exs1 13246 0csh0 13390 gcdcllem1 15059 nn0gsumfz 18203 pmatcollpw3lem 20407 pmatcollpw3fi1lem2 20411 pm2mpfo 20438 f1otrg 25551 cusgrafilem2 26008 wwlknredwwlkn 26254 wwlkextprop 26272 xrofsup 28923 esum2d 29482 rexzrexnn0 36386 ov2ssiunov2 37011 cusgrfilem2 40672 wwlksnredwwlkn 41101 wwlksnextprop 41118 clwwlksnun 41281 cusconngr 41358 lcoel0 42011 lcoss 42019 el0ldep 42049 ldepspr 42056 islindeps2 42066 isldepslvec2 42068 |
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