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Mirrors > Home > ILE Home > Th. List > spcv | GIF version |
Description: Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.) |
Ref | Expression |
---|---|
spcv.1 | ⊢ 𝐴 ∈ V |
spcv.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcv | ⊢ (∀𝑥𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcv.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | spcv.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
3 | 2 | spcgv 2824 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓)) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (∀𝑥𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 = wceq 1353 ∈ 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: morex 2921 exmidexmid 4193 exmidsssn 4199 exmidel 4202 rext 4211 ontr2exmid 4520 regexmidlem1 4528 reg2exmid 4531 relop 4772 disjxp1 6230 rdgtfr 6368 ssfiexmid 6869 domfiexmid 6871 diffitest 6880 findcard 6881 fiintim 6921 fisseneq 6924 finomni 7131 exmidomni 7133 exmidlpo 7134 exmidunben 12397 bj-d0clsepcl 14299 bj-inf2vnlem1 14344 subctctexmid 14373 |
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