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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 2891 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
2 | rspc2.1 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
3 | 1, 2 | nfralw 3298 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐷 𝜒 |
4 | rspc2.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
5 | 4 | ralbidv 3167 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 𝜒)) |
6 | 3, 5 | rspc 3595 | . 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 3595 | . 2 ⊢ (𝐵 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 𝜒 → 𝜓)) |
10 | 6, 9 | sylan9 506 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-v 3463 |
This theorem is referenced by: reu2eqd 3729 reuop 6303 fvmpocurryd 8285 dvmptfsum 25990 poimirlem26 37307 fphpd 42422 reupr 47043 |
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