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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 3900 | . . . . 5 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷} | |
| 2 | 1 | eqabri 2885 | . . . 4 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷) | 
| 3 | df-csb 3900 | . . . . . 6 ⊢ ⦋𝐵 / 𝑦⦌𝐷 = {𝑧 ∣ [𝐵 / 𝑦]𝑧 ∈ 𝐷} | |
| 4 | 3 | eqabri 2885 | . . . . 5 ⊢ (𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐵 / 𝑦]𝑧 ∈ 𝐷) | 
| 5 | 4 | sbcbii 3846 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ ⦋𝐵 / 𝑦⦌𝐷 ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷) | 
| 6 | 2, 5 | bitri 275 | . . 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 3900 | . . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐷} | |
| 11 | 10 | eqabri 2885 | . . . . 5 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐷 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐷) | 
| 12 | csbcom2fi.4 | . . . . . 6 ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸 | |
| 13 | 12 | eleq2i 2833 | . . . . 5 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌𝐷 ↔ 𝑧 ∈ 𝐸) | 
| 14 | 11, 13 | bitr3i 277 | . . . 4 ⊢ ([𝐴 / 𝑥]𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐸) | 
| 15 | 7, 8, 9, 14 | sbccom2fi 38134 | . . 3 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝑧 ∈ 𝐷 ↔ [𝐶 / 𝑦]𝑧 ∈ 𝐸) | 
| 16 | sbcel2 4418 | . . 3 ⊢ ([𝐶 / 𝑦]𝑧 ∈ 𝐸 ↔ 𝑧 ∈ ⦋𝐶 / 𝑦⦌𝐸) | |
| 17 | 6, 15, 16 | 3bitri 297 | . 2 ⊢ (𝑧 ∈ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 ↔ 𝑧 ∈ ⦋𝐶 / 𝑦⦌𝐸) | 
| 18 | 17 | eqriv 2734 | 1 ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 [wsbc 3788 ⦋csb 3899 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: (None) | 
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