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Metamath Proof Explorer

Theorem bnj1521

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

Ref Expression
Hypotheses bnj1521.1 ⊢ ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜑 )
bnj1521.2 ⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
bnj1521.3 ⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
Assertion bnj1521 ( 𝜒 → ∃ 𝑥 𝜃 )

Proof

Step Hyp Ref Expression
1 bnj1521.1 ⊢ ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜑 )
2 bnj1521.2 ⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
3 bnj1521.3 ⊢ ( 𝜒 → ∀ 𝑥 𝜒 )
4 1 bnj1196 ⊢ ( 𝜒 → ∃ 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) )
5 4 2 3 bnj1345 ⊢ ( 𝜒 → ∃ 𝑥 𝜃 )