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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 2977 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
2 | rspc2.1 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
3 | 1, 2 | nfralw 3225 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐷 𝜒 |
4 | rspc2.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
5 | 4 | ralbidv 3197 | . . 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 510 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 |
This theorem is referenced by: reu2eqd 3726 reuop 6138 fvmpocurryd 7931 dvmptfsum 24566 poimirlem26 34912 fphpd 39406 reupr 43678 |
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