Theorem bnj1441g

Description: First-order logic and set theory. See bnj1441 for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1441g.1	⊢ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 𝑥 ∈ 𝐴 )
	bnj1441g.2	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
Assertion	bnj1441g	⊢ ( 𝑧 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } → ∀ 𝑦 𝑧 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		bnj1441g.1	⊢ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 𝑥 ∈ 𝐴 )
2		bnj1441g.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	hbabg	⊢ ( 𝑧 ∈ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } → ∀ 𝑦 𝑧 ∈ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
6	3 5	hbxfreq	⊢ ( 𝑧 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } → ∀ 𝑦 𝑧 ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )