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Mirrors > Home > NFE Home > Th. List > rspcv | GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) |
Ref | Expression |
---|---|
rspcv.1 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rspcv | ⊢ (A ∈ B → (∀x ∈ B φ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1619 | . 2 ⊢ Ⅎxψ | |
2 | rspcv.1 | . 2 ⊢ (x = A → (φ ↔ ψ)) | |
3 | 1, 2 | rspc 2949 | 1 ⊢ (A ∈ B → (∀x ∈ B φ → ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 ∀wral 2614 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 |
This theorem is referenced by: rspccv 2952 rspcva 2953 rspccva 2954 rspc3v 2964 rr19.3v 2980 rr19.28v 2981 rspsbc 3124 intmin 3946 evenodddisj 4516 nnadjoin 4520 tfinnn 4534 spfinsfincl 4539 funcnvuni 5161 nnc3n3p1 6278 nchoicelem12 6300 nchoicelem19 6307 |
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