Theorem csbco 3114

Description: Composition law for chained substitutions into a class.

Use the weaker csbcow 3115 when possible. (Contributed by NM, 10-Nov-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
csbco	⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Distinct variable group:   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem csbco

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3105	. . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵}
2	1	abeq2i 2320	. . . . 5 ⊢ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵)
3	2	sbcbii 3068	. . . 4 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵)
4		sbcco 3030	. . . 4 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
5	3, 4	bitri 184	. . 3 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
6	5	abbii 2325	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵} = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
7		df-csb 3105	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵}
8		df-csb 3105	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
9	6, 7, 8	3eqtr4i 2240	1 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:   = wceq 1375   ∈ wcel 2180  {cab 2195  [wsbc 3008  ⦋csb 3104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-sbc 3009  df-csb 3105

This theorem is referenced by:  csbvarg  3132  csbnest1g  3160  zsumdc  11861  fsum3  11864  fsumsplitf  11885