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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 2764 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfv 1843 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | spcgv.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | spcgf 3288 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1481 = wceq 1483 ∈ wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 |
This theorem is referenced by: spcv 3299 mob2 3386 intss1 4492 dfiin2g 4553 alxfr 4878 fri 5076 isofrlem 6590 tfisi 7058 limomss 7070 nnlim 7078 f1oweALT 7152 pssnn 8178 findcard3 8203 ttukeylem1 9331 rami 15719 ramcl 15733 islbs3 19155 mplsubglem 19434 mpllsslem 19435 uniopn 20702 chlimi 28091 dfon2lem3 31690 dfon2lem8 31695 neificl 33549 ismrcd1 37261 |
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