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Metamath Proof Explorer

Theorem nfcnv

Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004) (Revised by Mario Carneiro, 15-Oct-2016)

Ref Expression
Hypothesis nfcnv.1 ⊢ Ⅎ 𝑥 𝐴
Assertion nfcnv Ⅎ 𝑥 ◡ 𝐴

Proof

Step Hyp Ref Expression
1 nfcnv.1 ⊢ Ⅎ 𝑥 𝐴
2 df-cnv ⊢ ◡ 𝐴 = { ⟨ 𝑦 , 𝑧 ⟩ ∣ 𝑧 𝐴 𝑦 }
3 nfcv ⊢ Ⅎ 𝑥 𝑧
4 nfcv ⊢ Ⅎ 𝑥 𝑦
5 3 1 4 nfbr ⊢ Ⅎ 𝑥 𝑧 𝐴 𝑦
6 5 nfopab ⊢ Ⅎ 𝑥 { ⟨ 𝑦 , 𝑧 ⟩ ∣ 𝑧 𝐴 𝑦 }
7 2 6 nfcxfr ⊢ Ⅎ 𝑥 ◡ 𝐴