Theorem csbco 3924

Description: Composition law for chained substitutions into a class. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker csbcow 3923 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 3909	. . . . . 6 ⊢ ⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐵}
2	1	eqabri 2883	. . . . 5 ⊢ (𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝐵)
3	2	sbcbii 3852	. . . 4 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵)
4		sbcco 3817	. . . 4 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝑧 ∈ 𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
5	3, 4	bitri 275	. . 3 ⊢ ([𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑧 ∈ 𝐵)
6	5	abbii 2807	. 2 ⊢ {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵} = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
7		df-csb 3909	. 2 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑦]𝑧 ∈ ⦋𝑦 / 𝑥⦌𝐵}
8		df-csb 3909	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑧 ∣ [𝐴 / 𝑥]𝑧 ∈ 𝐵}
9	6, 7, 8	3eqtr4i 2773	1 ⊢ ⦋𝐴 / 𝑦⦌⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:   = wceq 1537   ∈ wcel 2106  {cab 2712  [wsbc 3791  ⦋csb 3908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sbc 3792  df-csb 3909

This theorem is referenced by: (None)