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Metamath Proof Explorer

Theorem csbco

Description: Composition law for chained substitutions into a class. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker csbcow when possible. (Contributed by NM, 10-Nov-2005) (New usage is discouraged.)

Ref Expression
Assertion csbco ⊢ ⦋ 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 sbcco ⊢ [˙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