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Metamath Proof Explorer

Theorem csbnestg

Description: Nest the composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker csbnestgw when possible. (Contributed by NM, 23-Nov-2005) (Proof shortened by Mario Carneiro, 10-Nov-2016) (New usage is discouraged.)

Ref Expression
Assertion csbnestg ⊢ A ∈ V → ⦋ A / x⦌ ⦋ B / y⦌ C = ⦋ ⦋ A / x⦌ B / y⦌ C

Proof

Step Hyp Ref Expression
1 nfcv ⊢ Ⅎ _ x C
2 1 ax-gen ⊢ ∀ y Ⅎ _ x C
3 csbnestgf ⊢ A ∈ V ∧ ∀ y Ⅎ _ x C → ⦋ A / x⦌ ⦋ B / y⦌ C = ⦋ ⦋ A / x⦌ B / y⦌ C
4 2 3 mpan2 ⊢ A ∈ V → ⦋ A / x⦌ ⦋ B / y⦌ C = ⦋ ⦋ A / x⦌ B / y⦌ C