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Metamath Proof Explorer

Theorem bnj1345

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

Ref Expression
Hypotheses bnj1345.1 ⊢ ( 𝜑 → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )
bnj1345.2 ⊢ ( 𝜃 ↔ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
bnj1345.3 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
Assertion bnj1345 ( 𝜑 → ∃ 𝑥 𝜃 )

Proof

Step Hyp Ref Expression
1 bnj1345.1 ⊢ ( 𝜑 → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )
2 bnj1345.2 ⊢ ( 𝜃 ↔ ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
3 bnj1345.3 ⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4 1 3 bnj1275 ⊢ ( 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) )
5 4 2 bnj1198 ⊢ ( 𝜑 → ∃ 𝑥 𝜃 )