Theorem bnj1196 31720

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 3094	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	1, 2	sylib 210	1 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 387  ∃wex 1742   ∈ wcel 2050  ∃wrex 3089

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 199  df-rex 3094

This theorem is referenced by:  bnj1209  31722  bnj1265  31738  bnj1379  31756  bnj1521  31776  bnj900  31854  bnj986  31879  bnj1189  31932  bnj1245  31937  bnj1286  31942  bnj1311  31947  bnj1450  31973  bnj1498  31984