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

Metamath Proof Explorer

Theorem rabbid

Description: Version of rabbidv with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by BJ, 27-Apr-2019)

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

Proof

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