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Metamath Proof Explorer

Theorem hbxfrbi

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

	Ref	Expression
Hypotheses	hbxfrbi.1	⊢ ( 𝜑 ↔ 𝜓 )
	hbxfrbi.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
Assertion	hbxfrbi	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		hbxfrbi.1	⊢ ( 𝜑 ↔ 𝜓 )
2		hbxfrbi.2	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
3	1	albii	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑥 𝜓 )
4	2 1 3	3imtr4i	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )