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Metamath Proof Explorer

Theorem abbidv

Description: Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993) Avoid ax-12 , based on an idea of Steven Nguyen. (Revised by Wolf Lammen, 6-May-2023)

		Ref	Expression
	Hypothesis	abbidv.1	$⊢ φ \to (ψ \leftrightarrow χ)$
	Assertion	abbidv	$⊢ φ \to \{x \| ψ\} = \{x \| χ\}$

Proof

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