cbvab

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . Usage of the weaker cbvabw and cbvabv are preferred. (Contributed by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 16-Nov-2019) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbvab.1	$⊢ Ⅎ y φ$
	cbvab.2	$⊢ Ⅎ x ψ$
	cbvab.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
Assertion	cbvab	$⊢ \{x \| φ\} = \{y \| ψ\}$

Proof

Step	Hyp	Ref	Expression
1		cbvab.1	$⊢ Ⅎ y φ$
2		cbvab.2	$⊢ Ⅎ x ψ$
3		cbvab.3	$⊢ x = y \to (φ \leftrightarrow ψ)$
4	1	sbco2	$⊢ [z/ y] [y/ x] φ \leftrightarrow [z/ x] φ$
5	2 3	sbie	$⊢ [y/ x] φ \leftrightarrow ψ$
6	5	sbbii	$⊢ [z/ y] [y/ x] φ \leftrightarrow [z/ y] ψ$
7	4 6	bitr3i	$⊢ [z/ x] φ \leftrightarrow [z/ y] ψ$
8		df-clab	$⊢ z \in \{x \| φ\} \leftrightarrow [z/ x] φ$
9		df-clab	$⊢ z \in \{y \| ψ\} \leftrightarrow [z/ y] ψ$
10	7 8 9	3bitr4i	$⊢ z \in \{x \| φ\} \leftrightarrow z \in \{y \| ψ\}$
11	10	eqriv	$⊢ \{x \| φ\} = \{y \| ψ\}$

Metamath Proof Explorer

Theorem cbvab

Proof