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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 2976 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 | |
3 | rspc.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
4 | 2, 3 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
5 | eleq1 2899 | . . . . 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 3588 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
9 | 8 | anabsi5 667 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
10 | df-rex 3143 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
11 | 9, 10 | sylibr 236 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 Ⅎwnf 1783 ∈ wcel 2113 ∃wrex 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rex 3143 df-v 3493 |
This theorem is referenced by: rspcevOLD 3621 reuop 6137 ac6c4 9896 infcvgaux1i 15207 iunmbl2 24153 esumcvg 31366 ptrecube 34927 poimirlem24 34951 sdclem1 35051 uzwo4 41389 eliuniincex 41449 elrnmpt1sf 41524 iuneqfzuzlem 41676 uzublem 41778 uzub 41779 limsupubuzlem 42067 sge0gerp 42751 smflim 43127 reupr 43758 |
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