Theorem bnj1299 35215

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

Hypothesis

Ref	Expression
bnj1299.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
bnj1299	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Proof of Theorem bnj1299

Step	Hyp	Ref	Expression
1		bnj1299.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
2		bnj1239 35202	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 𝜓)
3	1, 2	syl 18	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-rex 3089

This theorem is used by:  bnj1497  35457  bnj1498  35458  bnj1501  35464