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Metamath Proof Explorer

Theorem csbco3g

Description: Composition of two class substitutions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 27-Nov-2005) (Revised by Mario Carneiro, 11-Nov-2016) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	sbcco3g.1	$⊢ x = A \to B = C$
	Assertion	csbco3g	$⊢ A \in V \to ⦋ A / x ⦌ ⦋ B / y ⦌ D = ⦋ C / y ⦌ D$

Proof

Step	Hyp	Ref	Expression
1		sbcco3g.1	$⊢ x = A \to B = C$
2		csbnestg	$⊢ A \in V \to ⦋ A / x ⦌ ⦋ B / y ⦌ D = ⦋ ⦋ A / x ⦌ B / y ⦌ D$
3		elex	$⊢ A \in V \to A \in V$
4		nfcvd	$⊢ A \in V \to \underline{Ⅎ} x C$
5	4 1	csbiegf	$⊢ A \in V \to ⦋ A / x ⦌ B = C$
6	3 5	syl	$⊢ A \in V \to ⦋ A / x ⦌ B = C$
7	6	csbeq1d	$⊢ A \in V \to ⦋ ⦋ A / x ⦌ B / y ⦌ D = ⦋ C / y ⦌ D$
8	2 7	eqtrd	$⊢ A \in V \to ⦋ A / x ⦌ ⦋ B / y ⦌ D = ⦋ C / y ⦌ D$