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Mirrors > Home > MPE Home > Th. List > spcgv | Structured version Visualization version GIF version |
Description: Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) |
Ref | Expression |
---|---|
spcgv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcgv | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1830 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | spcgv.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | spcgf 3261 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 |
This theorem is referenced by: spcv 3272 mob2 3353 intss1 4427 dfiin2g 4489 alxfr 4804 fri 5000 isofrlem 6490 tfisi 6950 limomss 6962 nnlim 6970 f1oweALT 7043 pssnn 8063 findcard3 8088 ttukeylem1 9214 rami 15557 ramcl 15571 islbs3 18976 mplsubglem 19255 mpllsslem 19256 uniopn 20527 0eusgraiff0rgra 26466 chlimi 27475 dfon2lem3 30934 dfon2lem8 30939 neificl 32719 ismrcd1 36279 |
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