Theorem csbco 3146

Description: Composition law for chained substitutions into a class. (Contributed by NM, 10-Nov-2005.)

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 3138	. . . . . 6 ⊢ [y / x]B = {z ∣ [̣y / x]̣z ∈ B}
2	1	abeq2i 2461	. . . . 5 ⊢ (z ∈ [y / x]B ↔ [̣y / x]̣z ∈ B)
3	2	sbcbii 3102	. . . 4 ⊢ ([̣A / y]̣z ∈ [y / x]B ↔ [̣A / y]̣[̣y / x]̣z ∈ B)
4		sbcco 3069	. . . 4 ⊢ ([̣A / y]̣[̣y / x]̣z ∈ B ↔ [̣A / x]̣z ∈ B)
5	3, 4	bitri 240	. . 3 ⊢ ([̣A / y]̣z ∈ [y / x]B ↔ [̣A / x]̣z ∈ B)
6	5	abbii 2466	. 2 ⊢ {z ∣ [̣A / y]̣z ∈ [y / x]B} = {z ∣ [̣A / x]̣z ∈ B}
7		df-csb 3138	. 2 ⊢ [A / y][y / x]B = {z ∣ [̣A / y]̣z ∈ [y / x]B}
8		df-csb 3138	. 2 ⊢ [A / x]B = {z ∣ [̣A / x]̣z ∈ B}
9	6, 7, 8	3eqtr4i 2383	1 ⊢ [A / y][y / x]B = [A / x]B

Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3047  [csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbvarg  3164  csbnest1g  3189  eqerlem  5961