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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.) Avoid ax-10 2138, ax-11 2155. (Revised by Wolf Lammen, 25-Aug-2023.) |
Ref | Expression |
---|---|
spcgv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spcgv | ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3465 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | id 22 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ V) | |
3 | spcgv.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 3 | adantl 483 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
5 | 2, 4 | spcdv 3555 | . 2 ⊢ (𝐴 ∈ V → (∀𝑥𝜑 → 𝜓)) |
6 | 1, 5 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 = wceq 1542 ∈ wcel 2107 Vcvv 3447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 |
This theorem is referenced by: spcv 3566 mob2 3677 sbceqal 3809 intss1 4928 dfiin2g 4996 alxfr 5366 friOLD 5598 funmo 6520 isofrlem 7289 tfisi 7799 limomss 7811 nnlim 7820 f1oweALT 7909 pssnn 9118 pssnnOLD 9215 findcard3 9235 findcard3OLD 9236 frmin 9693 ttukeylem1 10453 rami 16895 ramcl 16909 islbs3 20661 mplsubglem 21428 mpllsslem 21429 uniopn 22269 chlimi 30225 iinabrex 31540 dfon2lem3 34423 dfon2lem8 34428 neificl 36262 ismrcd1 41068 mnuop23d 42638 |
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