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Mirrors > Home > MPE Home > Th. List > rspc2vd | Structured version Visualization version GIF version |
Description: Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class 𝐷 for the second set variable 𝑦 may depend on the first set variable 𝑥. (Contributed by AV, 29-Mar-2021.) |
Ref | Expression |
---|---|
rspc2vd.a | ⊢ (𝑥 = 𝐴 → (𝜃 ↔ 𝜒)) |
rspc2vd.b | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) |
rspc2vd.c | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
rspc2vd.d | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐸) |
rspc2vd.e | ⊢ (𝜑 → 𝐵 ∈ 𝐸) |
Ref | Expression |
---|---|
rspc2vd | ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspc2vd.e | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝐸) | |
2 | rspc2vd.c | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
3 | rspc2vd.d | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐸) | |
4 | 2, 3 | csbied 3926 | . . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐷 = 𝐸) |
5 | 1, 4 | eleqtrrd 2830 | . 2 ⊢ (𝜑 → 𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐷) |
6 | nfcsb1v 3913 | . . . . 5 ⊢ Ⅎ𝑥⦋𝐴 / 𝑥⦌𝐷 | |
7 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑥𝜒 | |
8 | 6, 7 | nfralw 3302 | . . . 4 ⊢ Ⅎ𝑥∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒 |
9 | csbeq1a 3902 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝐷 = ⦋𝐴 / 𝑥⦌𝐷) | |
10 | rspc2vd.a | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜃 ↔ 𝜒)) | |
11 | 9, 10 | raleqbidv 3336 | . . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐷 𝜃 ↔ ∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒)) |
12 | 8, 11 | rspc 3594 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → ∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒)) |
13 | 2, 12 | syl 17 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → ∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒)) |
14 | rspc2vd.b | . . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) | |
15 | 14 | rspcv 3602 | . 2 ⊢ (𝐵 ∈ ⦋𝐴 / 𝑥⦌𝐷 → (∀𝑦 ∈ ⦋ 𝐴 / 𝑥⦌𝐷𝜒 → 𝜓)) |
16 | 5, 13, 15 | sylsyld 61 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜃 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⦋csb 3888 |
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 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-v 3470 df-sbc 3773 df-csb 3889 |
This theorem is referenced by: insubm 18743 frcond1 30028 frgrwopreglem4a 30072 ismntd 32659 dfmgc2lem 32670 urpropd 32882 isthincd2lem1 47918 |
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