New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > rspcimdv | GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rspcimdv.1 | ⊢ (φ → A ∈ B) |
rspcimdv.2 | ⊢ ((φ ∧ x = A) → (ψ → χ)) |
Ref | Expression |
---|---|
rspcimdv | ⊢ (φ → (∀x ∈ B ψ → χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2619 | . 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ)) | |
2 | rspcimdv.1 | . . 3 ⊢ (φ → A ∈ B) | |
3 | simpr 447 | . . . . . . 7 ⊢ ((φ ∧ x = A) → x = A) | |
4 | 3 | eleq1d 2419 | . . . . . 6 ⊢ ((φ ∧ x = A) → (x ∈ B ↔ A ∈ B)) |
5 | 4 | biimprd 214 | . . . . 5 ⊢ ((φ ∧ x = A) → (A ∈ B → x ∈ B)) |
6 | rspcimdv.2 | . . . . 5 ⊢ ((φ ∧ x = A) → (ψ → χ)) | |
7 | 5, 6 | imim12d 68 | . . . 4 ⊢ ((φ ∧ x = A) → ((x ∈ B → ψ) → (A ∈ B → χ))) |
8 | 2, 7 | spcimdv 2936 | . . 3 ⊢ (φ → (∀x(x ∈ B → ψ) → (A ∈ B → χ))) |
9 | 2, 8 | mpid 37 | . 2 ⊢ (φ → (∀x(x ∈ B → ψ) → χ)) |
10 | 1, 9 | syl5bi 208 | 1 ⊢ (φ → (∀x ∈ B ψ → χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wal 1540 = 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: rspcimedv 2957 rspcdv 2958 |
Copyright terms: Public domain | W3C validator |