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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 247 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓)) |
4 | 1, 3 | rspcimedv 3604 | 1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 |
This theorem is referenced by: rspcebdv 3607 rspcev 3613 rspcedvd 3615 0csh0 14740 gcdcllem1 16437 nn0gsumfz 19847 pmatcollpw3lem 22277 pmatcollpw3fi1lem2 22281 pm2mpfo 22308 f1otrg 28112 cusgrfilem2 28703 wwlksnredwwlkn 29139 wwlksnextprop 29156 clwwlknun 29355 cusconngr 29434 xrofsup 31968 esum2d 33080 rexzrexnn0 41528 onsucelab 41999 ordnexbtwnsuc 42003 ov2ssiunov2 42437 requad2 46278 lcoel0 47063 lcoss 47071 el0ldep 47101 ldepspr 47108 islindeps2 47118 isldepslvec2 47120 affinecomb1 47342 |
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