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Mirrors > Home > NFE Home > Th. List > rspc2 | GIF version |
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.) |
Ref | Expression |
---|---|
rspc2.1 | ⊢ Ⅎxχ |
rspc2.2 | ⊢ Ⅎyψ |
rspc2.3 | ⊢ (x = A → (φ ↔ χ)) |
rspc2.4 | ⊢ (y = B → (χ ↔ ψ)) |
Ref | Expression |
---|---|
rspc2 | ⊢ ((A ∈ C ∧ B ∈ D) → (∀x ∈ C ∀y ∈ D φ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2489 | . . . 4 ⊢ ℲxD | |
2 | rspc2.1 | . . . 4 ⊢ Ⅎxχ | |
3 | 1, 2 | nfral 2667 | . . 3 ⊢ Ⅎx∀y ∈ D χ |
4 | rspc2.3 | . . . 4 ⊢ (x = A → (φ ↔ χ)) | |
5 | 4 | ralbidv 2634 | . . 3 ⊢ (x = A → (∀y ∈ D φ ↔ ∀y ∈ D χ)) |
6 | 3, 5 | rspc 2949 | . 2 ⊢ (A ∈ C → (∀x ∈ C ∀y ∈ D φ → ∀y ∈ D χ)) |
7 | rspc2.2 | . . 3 ⊢ Ⅎyψ | |
8 | rspc2.4 | . . 3 ⊢ (y = B → (χ ↔ ψ)) | |
9 | 7, 8 | rspc 2949 | . 2 ⊢ (B ∈ D → (∀y ∈ D χ → ψ)) |
10 | 6, 9 | sylan9 638 | 1 ⊢ ((A ∈ C ∧ B ∈ D) → (∀x ∈ C ∀y ∈ D φ → ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 Ⅎwnf 1544 = 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: rspc2v 2961 |
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