Theorem bnj1441

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) Add disjoint variable condition to avoid ax-13 . See bnj1441g for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1441.1	⊢ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 𝑥 ∈ 𝐴 )
	bnj1441.2	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
Assertion	bnj1441	⊢ ( 𝑧 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } → ∀ 𝑦 𝑧 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		bnj1441.1	⊢ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 𝑥 ∈ 𝐴 )
2		bnj1441.2	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
3		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
4	1 2	hban	⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ∀ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
5	4	hbab	⊢ ( 𝑧 ∈ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } → ∀ 𝑦 𝑧 ∈ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
6	3 5	hbxfreq	⊢ ( 𝑧 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } → ∀ 𝑦 𝑧 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )