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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 2825 | . 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 2737 |
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 2739 |
This theorem is referenced by: bnd2 4173 mss 4226 exss 4227 snnex 4448 opeldm 4830 elrnmpt1 4878 xpmlem 5049 ffoss 5493 ssimaex 5577 fvelrn 5647 eufnfv 5747 foeqcnvco 5790 cnvoprab 6234 domtr 6784 ensn1 6795 ac6sfi 6897 difinfsn 7098 0ct 7105 ctmlemr 7106 ctssdclemn0 7108 ctssdclemr 7110 ctssdc 7111 omct 7115 ctssexmid 7147 exmidfodomrlemim 7199 cc3 7266 zfz1iso 10816 fprodntrivap 11587 ennnfonelemim 12419 ctinfom 12423 ctinf 12425 qnnen 12426 enctlem 12427 ctiunct 12435 nninfdc 12448 subctctexmid 14670 |
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