Theorem bnj1265 35442

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 35424	. . 3 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	bnj1266 35441	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
4	3	bnj937 35402	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-rex 3087

This theorem is used by:  bnj1253  35647  bnj1280  35650  bnj1296  35651  bnj1371  35659  bnj1497  35690