This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem abbid

Description: Equivalent wff's yield equal class abstractions (deduction form, with nonfreeness hypothesis). (Contributed by NM, 21-Jun-1993) (Revised by Mario Carneiro, 7-Oct-2016) Avoid ax-10 and ax-11 . (Revised by Wolf Lammen, 6-May-2023)

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

Proof

Step	Hyp	Ref	Expression
1		abbid.1	$⊢ Ⅎ x φ$
2		abbid.2	$⊢ φ \to (ψ \leftrightarrow χ)$
3	1 2	alrimi	$⊢ φ \to \forall x (ψ \leftrightarrow χ)$
4		abbi	$⊢ \forall x (ψ \leftrightarrow χ) \to \{x \| ψ\} = \{x \| χ\}$
5	3 4	syl	$⊢ φ \to \{x \| ψ\} = \{x \| χ\}$