Theorem bnj1196 30865

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

Hypothesis

Ref	Expression
bnj1196.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
bnj1196	⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))

Proof of Theorem bnj1196

Step	Hyp	Ref	Expression
1		bnj1196.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		df-rex 2918	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	1, 2	sylib 208	1 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-rex 2918

This theorem is referenced by:  bnj1209  30867  bnj1265  30883  bnj1379  30901  bnj1521  30921  bnj900  30999  bnj986  31024  bnj1189  31077  bnj1245  31082  bnj1286  31087  bnj1311  31092  bnj1450  31118  bnj1498  31129