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Metamath Proof Explorer

Theorem bnj1212

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1212.1 ⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 }
bnj1212.2 ⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏 ) )
Assertion bnj1212 ( 𝜃 → 𝑥 ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 bnj1212.1 ⊢ 𝐵 = { 𝑥 ∈ 𝐴 ∣ 𝜑 }
2 bnj1212.2 ⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜏 ) )
3 1 ssrab3 ⊢ 𝐵 ⊆ 𝐴
4 2 simp2bi ⊢ ( 𝜃 → 𝑥 ∈ 𝐵 )
5 3 4 bnj1213 ⊢ ( 𝜃 → 𝑥 ∈ 𝐴 )