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Mirrors > Home > MPE Home > Th. List > Mathboxes > csbcom2fi | Structured version Visualization version GIF version |
Description: Commutative law for double class substitution in a class, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.) |
Ref | Expression |
---|---|
csbcom2fi.1 | ⊢ 𝐴 ∈ V |
csbcom2fi.2 | ⊢ Ⅎ𝑦𝐴 |
csbcom2fi.3 | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 |
csbcom2fi.4 | ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸 |
Ref | Expression |
---|---|
csbcom2fi | ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3886 | . . . . 5 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷} | |
2 | 1 | abeq2i 2950 | . . . 4 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷) |
3 | df-csb 3886 | . . . . . 6 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐷} | |
4 | 3 | abeq2i 2950 | . . . . 5 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐷) |
5 | 4 | sbcbii 3831 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷) |
6 | 2, 5 | bitri 277 | . . 3 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷) |
7 | csbcom2fi.1 | . . . 4 ⊢ 𝐴 ∈ V | |
8 | csbcom2fi.2 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
9 | csbcom2fi.3 | . . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 | |
10 | df-csb 3886 | . . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐷} | |
11 | 10 | abeq2i 2950 | . . . . 5 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐷 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐷) |
12 | csbcom2fi.4 | . . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸 | |
13 | 12 | eleq2i 2906 | . . . . 5 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐷 ↔ 𝑧 ∈ 𝐸) |
14 | 11, 13 | bitr3i 279 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐸) |
15 | 7, 8, 9, 14 | sbccom2fi 35407 | . . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷 ↔ [𝐶 / 𝑦]𝑧 ∈ 𝐸) |
16 | sbcel2 4369 | . . 3 ⊢ ([𝐶 / 𝑦]𝑧 ∈ 𝐸 ↔ 𝑧 ∈ ⦋𝐶 / 𝑦⦌𝐸) | |
17 | 6, 15, 16 | 3bitri 299 | . 2 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ 𝑧 ∈ ⦋𝐶 / 𝑦⦌𝐸) |
18 | 17 | eqriv 2820 | 1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 Vcvv 3496 [wsbc 3774 ⦋csb 3885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-nul 4294 |
This theorem is referenced by: (None) |
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