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Metamath Proof Explorer

Theorem rabbida

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). Version of rabbidva with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019) Avoid ax-10 , ax-11 . (Revised by Wolf Lammen, 14-Mar-2025)

	Ref	Expression
Hypotheses	rabbida.n	⊢ Ⅎ 𝑥 𝜑
	rabbida.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	rabbida	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )

Proof

Step	Hyp	Ref	Expression
1		rabbida.n	⊢ Ⅎ 𝑥 𝜑
2		rabbida.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	pm5.32da	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝜒 ) ) )
4	1 3	rabbida4	⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )