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Mirrors > Home > ILE Home > Th. List > spcgf | GIF version |
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
spcgf.1 | ⊢ Ⅎ𝑥𝐴 |
spcgf.2 | ⊢ Ⅎ𝑥𝜓 |
spcgf.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcgf | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spcgf.2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | spcgf.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | 1, 2 | spcgft 2675 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓))) |
4 | spcgf.3 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | mpg 1380 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 Ⅎwnfc 2206 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 |
This theorem is referenced by: spcgv 2685 rspc 2695 elabgt 2735 eusvnf 4203 mpt2fvex 5849 |
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