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Metamath Proof Explorer

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 } )