Theorem bnj1196 32774

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 3070	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	1, 2	sylib 217	1 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065

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 206  df-rex 3070

This theorem is referenced by:  bnj1209  32776  bnj1265  32792  bnj1379  32810  bnj1521  32831  bnj900  32909  bnj986  32935  bnj1189  32989  bnj1245  32994  bnj1286  32999  bnj1311  33004  bnj1450  33030  bnj1498  33041