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

Proof

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