This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem csbcow

Description: Composition law for chained substitutions into a class. Version of csbco with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Nov-2005) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Assertion csbcow ⊢ ⦋ A / y⦌ ⦋ y / x⦌ B = ⦋ A / x⦌ B

Proof

Step Hyp Ref Expression
1 df-csb ⊢ ⦋ y / x⦌ B = z | [˙y / x]˙ z ∈ B
2 1 eqabri ⊢ z ∈ ⦋ y / x⦌ B ↔ [˙y / x]˙ z ∈ B
3 2 sbcbii ⊢ [˙A / y]˙ z ∈ ⦋ y / x⦌ B ↔ [˙A / y]˙ [˙y / x]˙ z ∈ B
4 sbccow ⊢ [˙A / y]˙ [˙y / x]˙ z ∈ B ↔ [˙A / x]˙ z ∈ B
5 3 4 bitri ⊢ [˙A / y]˙ z ∈ ⦋ y / x⦌ B ↔ [˙A / x]˙ z ∈ B
6 5 abbii ⊢ z | [˙A / y]˙ z ∈ ⦋ y / x⦌ B = z | [˙A / x]˙ z ∈ B
7 df-csb ⊢ ⦋ A / y⦌ ⦋ y / x⦌ B = z | [˙A / y]˙ z ∈ ⦋ y / x⦌ B
8 df-csb ⊢ ⦋ A / x⦌ B = z | [˙A / x]˙ z ∈ B
9 6 7 8 3eqtr4i ⊢ ⦋ A / y⦌ ⦋ y / x⦌ B = ⦋ A / x⦌ B