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Mirrors > Home > MPE Home > Th. List > csbcow | Structured version Visualization version GIF version |
Description: Composition law for chained substitutions into a class. Version of csbco 3892 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 10-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
csbcow | ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3877 | . . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵} | |
2 | 1 | abeq2i 2947 | . . . . 5 ⊢ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵) |
3 | 2 | sbcbii 3824 | . . . 4 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵) |
4 | sbccow 3791 | . . . 4 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵) | |
5 | 3, 4 | bitri 277 | . . 3 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵) |
6 | 5 | abbii 2885 | . 2 ⊢ {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵} = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} |
7 | df-csb 3877 | . 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵} | |
8 | df-csb 3877 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵} | |
9 | 6, 7, 8 | 3eqtr4i 2853 | 1 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 {cab 2798 [wsbc 3768 ⦋csb 3876 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1539 df-ex 1780 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-v 3493 df-sbc 3769 df-csb 3877 |
This theorem is referenced by: csbnest1g 4374 csbvarg 4376 fvmpocurryd 7930 zsum 15070 fsum 15072 fsumsplitf 15093 zprod 15286 fprod 15290 gsumply1eq 20468 f1od2 30457 bj-csbsn 34245 disjinfi 41528 climinf2mpt 42069 climinfmpt 42070 dvmptmulf 42296 dvmptfprod 42304 |
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