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Metamath Proof Explorer

Theorem rabbidv

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995)

Ref Expression
Hypothesis rabbidv.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion rabbidv ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )

Proof

Step Hyp Ref Expression
1 rabbidv.1 ⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2 1 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3 2 rabbidva ⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜓 } = { 𝑥 ∈ 𝐴 ∣ 𝜒 } )