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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 2955 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
2 | rspc2.1 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
3 | 1, 2 | nfralw 3189 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐷 𝜒 |
4 | rspc2.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
5 | 4 | ralbidv 3162 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 𝜒)) |
6 | 3, 5 | rspc 3559 | . 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 3559 | . 2 ⊢ (𝐵 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 𝜒 → 𝜓)) |
10 | 6, 9 | sylan9 511 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 |
This theorem is referenced by: reu2eqd 3675 reuop 6112 fvmpocurryd 7920 dvmptfsum 24578 poimirlem26 35083 fphpd 39757 reupr 44039 |
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