sbcco3g

Description: Composition of two substitutions. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker sbcco3gw when possible. (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	sbcco3g	$⊢ A \in V \to (\underset{˙}{[} A / x \underset{˙}{]} \underset{˙}{[} B / y \underset{˙}{]} φ \leftrightarrow \underset{˙}{[} C / y \underset{˙}{]} φ)$

Proof

Step	Hyp	Ref	Expression
1		sbcco3g.1	$⊢ x = A \to B = C$
2		sbcnestg	$⊢ A \in V \to (\underset{˙}{[} A / x \underset{˙}{]} \underset{˙}{[} B / y \underset{˙}{]} φ \leftrightarrow \underset{˙}{[} ⦋ A / x ⦌ B / y \underset{˙}{]} φ)$
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		dfsbcq	$⊢ ⦋ A / x ⦌ B = C \to (\underset{˙}{[} ⦋ A / x ⦌ B / y \underset{˙}{]} φ \leftrightarrow \underset{˙}{[} C / y \underset{˙}{]} φ)$
7	3 5 6	3syl	$⊢ A \in V \to (\underset{˙}{[} ⦋ A / x ⦌ B / y \underset{˙}{]} φ \leftrightarrow \underset{˙}{[} C / y \underset{˙}{]} φ)$
8	2 7	bitrd	$⊢ A \in V \to (\underset{˙}{[} A / x \underset{˙}{]} \underset{˙}{[} B / y \underset{˙}{]} φ \leftrightarrow \underset{˙}{[} C / y \underset{˙}{]} φ)$

Metamath Proof Explorer

Theorem sbcco3g

Proof