Theorem bnj1265 35209

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

Hypothesis

Ref	Expression
bnj1265.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
bnj1265	⊢ (𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem bnj1265

Step	Hyp	Ref	Expression
1		bnj1265.1	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2	1	bnj1196 35191	. . 3 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	bnj1266 35208	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
4	3	bnj937 35169	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142  ∃wrex 3088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-rex 3089

This theorem is used by:  bnj1253  35414  bnj1280  35417  bnj1296  35418  bnj1371  35426  bnj1497  35457