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| Mirrors > Home > MPE Home > Th. List > rspc2 | Structured version Visualization version GIF version | ||
| Description: Restricted specialization with two quantifiers, 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 2905 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
| 2 | rspc2.1 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
| 3 | 1, 2 | nfralw 3311 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐷 𝜒 |
| 4 | rspc2.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
| 5 | 4 | ralbidv 3178 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 𝜒)) |
| 6 | 3, 5 | rspc 3610 | . 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 3610 | . 2 ⊢ (𝐵 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 𝜒 → 𝜓)) |
| 10 | 6, 9 | sylan9 507 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 |
| This theorem is referenced by: reu2eqd 3742 reuop 6313 fvmpocurryd 8296 dvmptfsum 26013 poimirlem26 37653 fphpd 42827 reupr 47509 |
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