Theorem bnj1196 34784

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 3054	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	1, 2	sylib 218	1 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053

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 207  df-rex 3054

This theorem is referenced by:  bnj1209  34786  bnj1265  34802  bnj1379  34820  bnj1521  34841  bnj900  34919  bnj986  34945  bnj1189  34999  bnj1245  35004  bnj1286  35009  bnj1311  35014  bnj1450  35040  bnj1498  35051