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Metamath Proof Explorer

Theorem bnj1340

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

Ref Expression
Hypotheses bnj1340.1 ⊢ ( 𝜓 → ∃ 𝑥 𝜃 )
bnj1340.2 ⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝜃 ) )
bnj1340.3 ⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
Assertion bnj1340 ( 𝜓 → ∃ 𝑥 𝜒 )

Proof

Step Hyp Ref Expression
1 bnj1340.1 ⊢ ( 𝜓 → ∃ 𝑥 𝜃 )
2 bnj1340.2 ⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝜃 ) )
3 bnj1340.3 ⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
4 3 1 bnj596 ⊢ ( 𝜓 → ∃ 𝑥 ( 𝜓 ∧ 𝜃 ) )
5 4 2 bnj1198 ⊢ ( 𝜓 → ∃ 𝑥 𝜒 )