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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 → ( 𝜓 ↔ 𝜒 ) )
Assertion	abbid	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		abbid.1	⊢ Ⅎ 𝑥 𝜑
2		abbid.2	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
3	1 2	alrimi	⊢ ( 𝜑 → ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) )
4		abbi	⊢ ( ∀ 𝑥 ( 𝜓 ↔ 𝜒 ) → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ 𝜒 } )
5	3 4	syl	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ 𝜒 } )