Theorem bnj1265 32771

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 32753	. . 3 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
3	2	bnj1266 32770	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
4	3	bnj937 32730	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2109  ∃wrex 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1786  df-rex 3071

This theorem is referenced by:  bnj1253  32976  bnj1280  32979  bnj1296  32980  bnj1371  32988  bnj1497  33019