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Mirrors > Home > NFE Home > Th. List > rspccv | GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.) |
Ref | Expression |
---|---|
rspcv.1 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rspccv | ⊢ (∀x ∈ B φ → (A ∈ B → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcv.1 | . . 3 ⊢ (x = A → (φ ↔ ψ)) | |
2 | 1 | rspcv 2951 | . 2 ⊢ (A ∈ B → (∀x ∈ B φ → ψ)) |
3 | 2 | com12 27 | 1 ⊢ (∀x ∈ B φ → (A ∈ B → 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-mp 5 ax-1 6 ax-2 7 ax-3 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: elinti 3935 pw1disj 4167 peano2 4403 nndisjeq 4429 nnadjoin 4520 tfinnn 4534 fvun1 5379 refd 5927 nclenn 6249 nnc3n3p1 6278 nchoicelem12 6300 nchoicelem17 6305 |
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