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Metamath Proof Explorer

Theorem hbalw

Description: Weak version of hbal . Uses only Tarski's FOL axiom schemes. Unlike hbal , this theorem requires that x and y be distinct, i.e., not be bundled. (Contributed by NM, 19-Apr-2017)

	Ref	Expression
Hypotheses	hbalw.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
	hbalw.2	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
Assertion	hbalw	⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		hbalw.1	⊢ ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) )
2		hbalw.2	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
3	2	alimi	⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
4	1	alcomimw	⊢ ( ∀ 𝑦 ∀ 𝑥 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )
5	3 4	syl	⊢ ( ∀ 𝑦 𝜑 → ∀ 𝑥 ∀ 𝑦 𝜑 )