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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 3493 | . 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 3585 | . 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 3475 |
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 3477 |
This theorem is referenced by: spcv 3596 mob2 3712 sbceqal 3844 intss1 4968 dfiin2g 5036 alxfr 5406 friOLD 5638 funmo 6564 isofrlem 7337 tfisi 7848 limomss 7860 nnlim 7869 f1oweALT 7959 pssnn 9168 pssnnOLD 9265 findcard3 9285 findcard3OLD 9286 frmin 9744 ttukeylem1 10504 rami 16948 ramcl 16962 islbs3 20768 mplsubglem 21558 mpllsslem 21559 uniopn 22399 chlimi 30518 iinabrex 31831 dfon2lem3 34788 dfon2lem8 34793 neificl 36669 ismrcd1 41484 mnuop23d 43073 |
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