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Metamath Proof Explorer

Theorem hblem

Description: Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993) (Revised by Andrew Salmon, 11-Jul-2011) Add disjoint variable condition to avoid ax-13 . See hblemg for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

		Ref	Expression
	Hypothesis	hblem.1	⊢ ( 𝑦 ∈ 𝐴 → ∀ 𝑥 𝑦 ∈ 𝐴 )
	Assertion	hblem	⊢ ( 𝑧 ∈ 𝐴 → ∀ 𝑥 𝑧 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		hblem.1	⊢ ( 𝑦 ∈ 𝐴 → ∀ 𝑥 𝑦 ∈ 𝐴 )
2	1	hbsbw	⊢ ( [ 𝑧 / 𝑦 ] 𝑦 ∈ 𝐴 → ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝑦 ∈ 𝐴 )
3		clelsb1	⊢ ( [ 𝑧 / 𝑦 ] 𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴 )
4	3	albii	⊢ ( ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝑦 ∈ 𝐴 ↔ ∀ 𝑥 𝑧 ∈ 𝐴 )
5	2 3 4	3imtr3i	⊢ ( 𝑧 ∈ 𝐴 → ∀ 𝑥 𝑧 ∈ 𝐴 )