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Metamath Proof Explorer

Theorem csbnestgw

Description: Nest the composition of two substitutions. Version of csbnestg with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 23-Nov-2005) Avoid ax-13 . (Revised by GG, 26-Jan-2024)

Ref Expression
Assertion csbnestgw ⊢ 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 csbnestgfw ⊢ 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