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Mirrors > Home > ILE Home > Th. List > spcev | GIF version |
Description: Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
Ref | Expression |
---|---|
spcv.1 | ⊢ 𝐴 ∈ V |
spcv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcev | ⊢ (𝜓 → ∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcv.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | spcv.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | spcegv 2826 | . 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (𝜓 → ∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 Vcvv 2738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2740 |
This theorem is referenced by: bnd2 4174 mss 4227 exss 4228 snnex 4449 opeldm 4831 elrnmpt1 4879 xpmlem 5050 ffoss 5494 ssimaex 5578 fvelrn 5648 eufnfv 5748 foeqcnvco 5791 cnvoprab 6235 domtr 6785 ensn1 6796 ac6sfi 6898 difinfsn 7099 0ct 7106 ctmlemr 7107 ctssdclemn0 7109 ctssdclemr 7111 ctssdc 7112 omct 7116 ctssexmid 7148 exmidfodomrlemim 7200 cc3 7267 zfz1iso 10821 fprodntrivap 11592 ennnfonelemim 12425 ctinfom 12429 ctinf 12431 qnnen 12432 enctlem 12433 ctiunct 12441 nninfdc 12454 subctctexmid 14753 |
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