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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 250 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) |
4 | 1, 3 | rspcimedv 3606 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2814 df-clel 2893 df-ral 3143 df-rex 3144 |
This theorem is referenced by: rspcebdv 3609 rspcev 3615 rspcedvd 3618 0csh0 14140 gcdcllem1 15831 nn0gsumfz 19087 pmatcollpw3lem 21374 pmatcollpw3fi1lem2 21378 pm2mpfo 21405 f1otrg 26643 cusgrfilem2 27224 wwlksnredwwlkn 27659 wwlksnextprop 27676 clwwlknun 27875 cusconngr 27954 xrofsup 30478 esum2d 31359 rexzrexnn0 39493 ov2ssiunov2 40135 requad2 43873 lcoel0 44568 lcoss 44576 el0ldep 44606 ldepspr 44613 islindeps2 44623 isldepslvec2 44625 affinecomb1 44774 |
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