Theorem bnj1196 35290

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 3089	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	1, 2	sylib 221	1 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088

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

This proof depends on definitions:  df-bi 210  df-rex 3089

This theorem is used by:  bnj1209  35292  bnj1265  35308  bnj1379  35326  bnj1521  35347  bnj900  35425  bnj986  35451  bnj1189  35505  bnj1245  35510  bnj1286  35515  bnj1311  35520  bnj1450  35546  bnj1498  35557