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Metamath Proof Explorer

Theorem nfcxfrd

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016)

Ref Expression
Hypotheses nfcxfr.1 ⊢ 𝐴 = 𝐵
nfcxfrd.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
Assertion nfcxfrd ( 𝜑 → Ⅎ 𝑥 𝐴 )

Proof

Step Hyp Ref Expression
1 nfcxfr.1 ⊢ 𝐴 = 𝐵
2 nfcxfrd.2 ⊢ ( 𝜑 → Ⅎ 𝑥 𝐵 )
3 1 nfceqi ⊢ ( Ⅎ 𝑥 𝐴 ↔ Ⅎ 𝑥 𝐵 )
4 2 3 sylibr ⊢ ( 𝜑 → Ⅎ 𝑥 𝐴 )