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Mirrors > Home > MPE Home > Th. List > rspce | Structured version Visualization version GIF version |
Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.) |
Ref | Expression |
---|---|
rspc.1 | ⊢ Ⅎ𝑥𝜓 |
rspc.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspce | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | |
3 | rspc.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfan 1897 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
5 | eleq1 2827 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | rspc.2 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
8 | 1, 4, 7 | spcegf 3592 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
9 | 8 | anabsi5 669 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
10 | df-rex 3069 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
11 | 9, 10 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2106 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rex 3069 |
This theorem is referenced by: reuop 6315 ac6c4 10519 infcvgaux1i 15890 iunmbl2 25606 gsumpart 33043 esumcvg 34067 ptrecube 37607 poimirlem24 37631 sdclem1 37730 uzwo4 44993 eliuniincex 45049 elrnmpt1sf 45132 iuneqfzuzlem 45284 uzublem 45380 uzub 45381 limsupubuzlem 45668 sge0gerp 46351 smflim 46733 reupr 47447 |
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